Is speed of propeller-driven airplane limited by the speed of sound?

Don't forget that the aeroplane will be moving forward, so it's not relying on a vacuum filling ahead of the propellor to supply the latter with air.

Now I daresay there are good engineering reasons why propellors are not efficient and even impracticable for supersonic flight, but I don't think there is a fundamental physics theoretical reason ruling them out.

A propellor, from a theoretical standpoint, is not greatly different from a gas turbine jet or even a rocket insofar that it is simply "throwing stuff backwards", thus thrusting off the air it throws and being pushed forward by dint of Newton's third law. If it can be supplied enough air to throw backwards (and I think my first sentence shows there is probably no shortage of supply) and if it can impart a high enough impulse to the air, then there is no in principle limit to how fast the air is thrown backwards by the propellor. What happens if it is thrown backwards at faster than the speed of sound? Well in this case there would be an overpressure, meaning there would be a buildup of air there, the air thus becomes denser and "stiffer", and the local speed of sound behind the propellor can thus be much higher that that of the surrounding air. As the propellor does this, the air will undergo a sharp adiabatic temperature rise. Carrying this idea to its logical extreme, observe that rocket engines throw gas out behind them at roughly 10 times the speed of sound. It's simply a matter of how much you accelerate the gas - in principle there's no difference whether this acceleration is achieved by chemical energy or a great big bat whacking the air backwards.

Not physically, but practically there are (currently) better alternatives.

The limiting issue with propellers is similar to the limiting issue with helicopters: propellers work like wing sections in that they must accelerate flow to work; when you're near the speed of sound, this means you are going to cause shocks to form, and this issue is particularly bad (compared to a turbofan for example) as propellers work by accelerating a large mass of flow by a little, which means you make your propellers very large, and by extension, the length of your shock increases; shocks damage everything and they also require a vast quantity of energy to overcome - aerodynamicists like to avoid shocks.

Edit: The missing piece of information here is that whilst many people understand that, in a 1D tube, flow accelerates as cross sectional area decreases, up to M=1 (conservation of mass) - what is less well known is that as the cross sectional area then increases, the flow can continue to accelerate to M>1. This is the result of compressibility, amongst other things. Because of this there is no physical limit (insert some hand waving here, one certainly exists it's just not applicable in this region) we cannot exceed when accelerating our flow with a wing section e.g. a prop. The thing is that as we always slowly lose energy from our flow, at some point a supersonic flow will always return to subsonic conditions, which happens with a violent strong shock, across which an awful lot of energy is converted from K.E to heat.

As a result of improving understanding in this area, expect transonic aircraft in the next few decades to transition to turboprop designs with counter-rotating props.

A propeller can work at supersonic speeds because as it approaches those speeds it is catching up with the air molecules as it moves. So you don't have to "wait" for the molecules to move into the vacuum you create.

In other words the thrust of a propeller does not go to zero just because the plane reaches a certain speed.

But it is not enough to have thrust. You need "lots of" thrust...