I think there's a misunderstanding about space going on here. According to the title, you want to get an object into orbit with a cannon of some kind. This can't be done; every orbit includes the point at which delta-V was last spent. If you're launching from some kind of cannon, that means the orbit intersects with that cannon and probably the ground. Rockets in real life get around this by burning their engines again once they're way above the atmosphere.

But you don't need to get an object into orbit to hit something in orbit. You just need to reach the same altitude, which is much much easier. I think a volcano is your best bet; a supervolcanic eruption might be able to launch a rock into space. As I said, not into orbit though.

You may choose a different scenario.

Start with a supervolcano eruption. No, not one to hit the space station. One that will throw enough material into the atmosphere that it blocks most if not all radio-magnetic transmission. The dust in the atmosphere will last for months, even years, so you have plenty of time to get rid of humans while they are cut from their communication and then being invisible from the orbit prepare something to destroy the station in not-so-natural way. Another benefit is that this can change your planet's climate in a way that can itself be deadly at least to some level to the intruders.

Not a chance.

1. 
   

   You need to give the projectile all of the energy it needs in a very brief space of time (the time it needs to clear the volcano, to mention an example). That is an awesome amount of power. Remember that our rockets carry their fuel with them and burn it as they raise, it is highly inefficient but the alternative method would destroy the rocket at launch.

   
   
   

   You need some sturdy material for your projectile, and worse, you need to reinforce a lot the structure of any volcano, geyser, whatever that you want to use as your energy source. And not only the "cannon", but all around it, because you cannot allow that pressure that you need to build to escape by blowing some other part of the volcano.

   
   
   

   Mind you, completing engineering works of this scale in an active volcano is not an easy task, and likely to be observed by the ship. Overcoming this hurdle is a really unbelievable feat.

   
   



2. Precision. The worst one. You are sending an unguided projectile towards a minuscule target in a big sky. Any mistake in the measurements of the target orbit, your projectile weight and aerodinamics, your energy source power, the local weather, and you have nothing but a piece of rock wandering around your planet, almost harmless. Chaos theory says that you will fail.




3. Countermesures. It is difficult to believe that a spaceship by an advanced spacefaring civilization will not have some means of detecting and avoiding a meteorite strike (at the very least just changing its speed to avoid the crash).

Can you manipulate your local equivalent of the Van Allen belts to focus solar radiation and fry the satellite's electronics? Gets rid of the off-world communications and the video link without actually destroying the satellite..

On earth, the most common natural looking formation that can shoot rocks is a volcano. Yes, we know that a volcano shoots rocks high above the earth but not outside the atmosphere. So we can just modify the volcano to make the crater thinner, small and more round so that there is perfection in the accuracy. For launching a rock into the orbit we need the rock to be in an aerodynamic shape and have a highly flammable material on the bottom of the rock which can be triggered by the hot lava. So we have a volcano shooting out a rock missile into orbit. But we still won't have the right pressure so we will try to pressurize the volcano from within. We can use a system that lets lava come in but not out so that we have enough pressure and we can use a big door that will be opened when the pressure is right.

This might work but the accuracy will not be so good. So you can just launch it in such an orbit that it will crash with the station in the orbit and not directly hit it.

OP you specified earth-like but this is usually used to define atmosphere makeup and temperatures - what about a smaller planet (lower escape velocity), natural geologic formations that are much taller than the ones on earth due to tectonic activity? One could create something that is essentially a space-sling from a spot that is much higher altitude and solves most of your dilemmas. People on the ground might not even notice that there is something in the slingshot, which would be moving at a much higher velocity and could be released with pinpoint accuracy during the normal rotation of the earth.

Hell, you could have some kind of natural vine, native to the planet, that 'unintentionally' created a space elevator that the parasites utilized to ship materials to and from the station at one time. Then one day, when the planet's spin managed to line up with the station, suddenly let out a lot of extra line, swinging the end of the elevator in to the blister and destroying it, then retracting it immediately. No muss, no fuss.

Have you considered putting the orbiting station in a low orbit that needs occasional re-boosting to stop it de-orbiting? Spy satellites are often in low orbits to get the best detail in their pictures. The International Space Station would crash after about 15 months if abandoned.

If you can knock out the communications and any automated re-boosting mechanism, the satellite would burn up and take all the evidence of what happened with it so nobody can tell the difference between a software bug, a manufacturing defect, and a meteoroid strike.

Then if the gun from LSerni's great answer could fire something as radioactive as the heart of the Chernobyl sarcophagus, you could claim it only needed to get within a few tens of meters. All the humans see is their satellite has stopped responding and they can't tell why, and it burns up on re-entry before anyone can get there to investigate.

1. If you want to destroy their station, set a cloud of gas on the orbit (if it's not very high). You will get the most part of your gas back later. Do it several times while the station is on the other side of the planet. The speed of the station will decrease and it will get down. It will get damaged at first. Repeat while they not repair it.




2. Exotic things like atmospheric gas laser, microgravity resonance etc are not practical.



3. 
   

   Use your strength. Yours is time (on geological scales) Theirs is money.

   
   
   
   
   

   • Methodically destroy every habitat ship landing. They will find correlation very soon.

   
   

   • They will use robots. Destroy power plants.

   
   

   • They will use nano-robots. Destroy stocks and launch sites to prevent them stealing your resources.

   
   

   • They will use molecule-level robots to set you to a gray goo. Develop this technology

   
   
   
   



4. Those monsters will come back. They will set more than 1 station quite far from the planet. So get ready. The maximum reasonable impact is asteroids they will set on you. You are not scared of atomic bombs, aren't you? Well, they would set a mini black hole or something like this to finally get energy from you but I cannot get a solution for this case now.




5. Use energy of your star to get out of the habitat zone in your planet system. May be it would be even profitable for you, you just did not think about it. It will take a lot of time so start right now. If you use some space inside the planet it probably is not critical for you.




6. Split and diversify your bodies for different conditions, impacts and possibilities. This incident will be start of your infinite expansion to the space. They are split by their greed. You will get unite with the new infinite goal that will transform you. Set yourself total on this. They got here to wake up you. This is the aim of any real contact. Get their passion and apply with your planet-wide scale. Make all possible for this, not to get rid of that piece of shit on the orbit.

@Chris Bradshaw presented an elegant solution for big distances, but the Juno Radiation Vault mission proves that even today's shielded electronics can work some years even in radiation belts of Jupiter, which are more intensive than ones for an Earth-like planet

Probably the author will have a strike on the ground. He definitely will win the first one and will have some time to another. Any solutions to rise costs of taking ownership on the planet to unacceptable, including terraforming to temperature and atmospheric conditions unacceptable for invaders. Like on Venus. Oh, mankind is a big pain in the ass.

But wait a minute. "These humans have decided to colonize my home, mistaking it for an uninhabited Earth-like planet."

Are we talking about Earth? Because our thoughts more likely are inspired by our environment, not by something else like Solaris in hundreds of light years somewhere.

The space station is the opposite side for the action. As the center of crazy mind like a Death Star. They think they are only rational, legitimate, self-consistent and independent in all the space around.

Assuming an earth-like planet...

Satellites are not meant to land

Strategic heating of the atmosphere combined with a well placed hurricane or five and you can raise the atmosphere to the point where if they aren't watching they might accidentally enter the atmosphere, burn up and crash. As is the ISS actually experiences significant atmospheric drag ramping this up a bit will significantly lessen the life span of satellites in low orbit.

The Good

Changes to surface albedo, minor core cooling, tropospheric disturbances cause no long term losses of mass. It looks completely like a fluke.

The Bad

Anything designed for interstellar travel shouldn't have problem station keeping indefinitely.

The Fix

The observation post was intended to be temporary. It was put in an orbit that would only last 50 years before atmospheric entry. Low so that it could cover more ground faster, but not so low it would need station keeping fuel and thrusters.

Radiation Squeeze

The hypothetical planet has a magnetic field which blocks incoming radiation and ends up trapping it in high orbit much like our Van Allen Radiation belts. These ions are mostly free protons and electrons in belts segregated by charge. There are also limited amounts of positrons and anti-electrons as well as ionized helium floating around. The particles can destroy equipment and will certainly disrupt electronics. If you adjust the magnetic field of the planet to bring the belt into path of the satellite it may degrade its performance to the point where it no longer functions.

The Good

No mass lost. The South Atlantic Anomaly (where the belt reaches 200km above the surface as opposed to 2000km) is a thing so this looks like an unhappy planet to be.

The Bad

Interplanetary craft must deal with solar flares which can be far worse.

The Fix

Protecting the electronics from radiation surges often means shutting them down. If the shutdown needs to be longer than expected or more often than expected the station may be damaged anyway. Time this with a strong solar flare and you might be able to take it down.

Both at once

The planet is looking nice and hospitable. Strong spherical magnetic field nicely aligned with it smooth regular surface. Even flow of a liquid core. It's a relatively cool planet without strong temperature gradients. The sun is entering a maximum of activity but it shouldn't matter, everything is okay inside of the planets magnetic belt. Geologists are reporting that a large glacial lake is about to rupture some time in the next thousand years and it should be a spectacular display.

The earthen dam gave way far earlier than expected but such events have never been observed live. The biggest change and a large layer of fresh water on top of the planets largest ocean. In other notes there is a small magnetic anomaly forming at mid latitudes. There is no projected change to the magnetosphere at this time, but watch for variations in local declination.

Meteorology reports that turbidity is increasing and sea surface temperature is rising. A small dust storm is forming in the basin left by the drained glacier. They expect that a hurricane will be forming in the next 3-5 days and rapidly move to mid latitudes. The first one observed on this planet.

The hurricane has formed. Meteorology is going wild. There hasn't been a storm this big recorded ever on earth. Solar flare predicted to impact in 2 days Observation station going into protective shut down mode. The magnetic anomaly has strengthened. It looks like we are at the beginning of a shift of the planets magnetic poles.

The solar flare was stronger and faster than expected the system only got partially shut down and there are some data corruption issues. The information we are getting is only intermittent. We are attempting a reboot from the ground station.

More data is in coming in now. Full system integrity checks take two hours to complete. There is some sort of radiation anomaly occurring forcing a reset every 93 minutes. Atmospheric craft report the hurricane has only intensified.

Communication has been fully restored with with observation station. It seems the planets Van Allen belts have reoriented themselves with the magnetic poles shift. In order to maintain the observation stations integrity we need to shut it down when passing through the anomaly. Unfortunately, the anomaly occurs near apogee. We are trying to schedule a station keeping maneuver but the orbit is decaying far faster then predicted.

1. What is the most energetic phenomenon a planet might have access to? I here assert this to be impacts from extraplanetary objects.




2. Could an impact from an extraplanetary object propel material off of the impacted planet fast enough to escape the gravity well of the planet? Yes - a meteorite loosed from Mars was in the news some years ago as potentially containing fossil microbes. An ancient rock blasted loose from Earth was recently discovered in a collection of moon rocks.




3. Could rocks blasted loose from the planet by an impactor disrupt an orbiting satellite? I think yes. Clearly a rock moving fast enough to reach the moon could do some damage if it hit something en route. I would assume this moon-destined rock was one of many and that the impact would produce a shotgun-like giant cloud of rock.




4. Could a planetary super creature induce an impactor to hit? This is where it gets tricky. Let us assume the creature can manipulate the planet's magnetic field by altering magmatic flow. A magnetic field can induce a current within a conductor moving within it; metal asteroids / meteors would be conductors. The current within the moving asteroid generates heat. The energy to create this heat is taken from the kinetic energy of the moving asteroid thru the magnetic field, thus slowing the asteroid. This slowing /heating would occur for a meteor moving perpendicular to lines of force, not parallel. http://adsabs.harvard.edu/full/1946PA.....54..482R

Thus by manipulating magnetic fields, the creature could steer a meteor to some degree, possibly inducing a meteor to hit in such a way as to launch a cloud of rock at escape velocity, obliterating the satellite.

This induced mega impact is kind of a bold maneuver given the creature calls the planet home. But if it can steer nearby asteroids to some degree, why not steer a small one into the spacecraft and leave out the massive impact part? Even this would take great effort on the part of the creature as well as knowledge of asteroids in the vicinity (which it might perceive, at least periodically, if it were an electromagnetic creature).

It might miss the first several times it tried. In fact for the story I like the idea of it missing every time but then realizing that the alteration of the magnetic field actually was greatly altering the stream of charged particles from the solar wind. Rather than whack the satellite with a fast moving piece of metal, it rearranges the planet's field so charged particles are funneled down onto the satellite, producing a quiet death from a thousand cuts.