The vacuum chamber on the end of a rocket is a really cool idea! Lets test it:

Assumptions:
Vacuum chamber is huge (lets just say many miles long) and add no mass or air resistance.
Exhaust from rocket only goes into vacuum chamber.

What happens?
Chemicals (liquid hydrogen and oxygen) react making a very hot gas. In a hot gas the molecules move very fast and shoot out the nozzle. Due to conservation of momentum the molecules move one way then the rocket will move the other way.

Momentum always is conserved:
total momentum before: 0 - both the fuel and rocket are at rest
total momentum after: 0 - hot gas shooting one way, rocket shoots other way

The hot gas doesn't need to push off of anything, its all just conservation of momentum like recoil from a gun or throwing a ball while ice skating.

HOWEVER: The rocket will eventually slow down when the hot gas hits the back side of the vacuum chamber, but that besides the point since there is no back side to space.

Granted this is all based on the ASSUMPTION that momentum is conserved. Yes for everyone out there I said ASSUMPTION. Science is a constantly evolving field. Everything including rockets that we have observed obeys conservation of momentum. One day we may find some situation/object that doesn't, but until that day I CHOOSE to believe it. And if you choose no to believe in conservation of momentum, then that is FINE, but you should have a good reason or explanation for everything that we observe that says conservation of momentum is valid.

Also, I work at x-ray rocket lab. Yes we dont go into deep space, but then again NONE of space is truely a vacuum. Anyway, we have to test our rockets in a vacuum chamber to make sure they will work. It take a lot to hold the rockets in place because when fired they exert a force. if a rocket didn't work in space you could just set them on a table and the wouldn't go anywhere and my job would be alot easier