Satellites only need sufficient motion and gravity. They've orbited things that aren't quite round/spherical (asteroids and comets), and theoretically flat things too, if we can find/build one big enough.
Orbits of nonspherical objects tend to be unstable (this is a significant effect for lunar orbits already) and the motion of the satellites would be completely different.
If a flat earth existed, our notions of gravity are definitely wrong, because a flat earth would probably quickly collapse into a lump- and the direction of gravity would point in a different direction as you walked towards the edge...
There is a video about this which is quite funny. You wouldn't actually fall of the edge but could stand on it comfortably.
VSauce made such a video on this topic. Quite interesting
Their explanation for Earth's gravity is that the Earth is constantly accelerating "upward" at 8.9m/s/s. How would satellites orbit that without constant acceleration of their own?