Yes the speed of a propeller driven aircraft is limited by the speed of sound, but not in the way you think. What's happening:
The propeller aircraft has a speed of its own and flies into a bunch of previously unsuspecting air, which was twiddling its thumbs. At subsonic aircraft speeds, the air in front of the plane gets a warning, and starts to assemble itself towards the propellor.
The incoming air hits the propeller disc at a higher total pressure than ambient pressure - no vacuum here! And the propeller imparts energy onto the incoming air stream, and accelerates it backwards. This provides the thrust that enables the aircraft to fly.
The picture above explains the principle of thrust generation using the momentum method, which is useful but does not concern itself with the power required to turn the propeller. The propeller blades are miniature wings spinning around, providing thrust (similar to lift in a wing) and requiring torque (similar to drag in a wing). The blade tip speed is the vector sum of rotational velocity and airspeed.
And if the tip speed becomes supersonic, that is where the limiting factor is. At supersonic speed, the shock waves produce an extraordinary amount of drag without any useful increase in lift whatsoever, so a rapid increase in required engine power without the propeller producing more thrust.
Note that the limiting speed is the propeller tip speed. The aircraft air speed is one of the vectors contributing to this, and is therefore always lower than the speed of sound. Practical limits for propeller aircraft are around M=0.6, above which jet engines are required.