Would rockets work in a vacuum?

Yes they do.

The rockets fuel, either solid, compressed gas would expand on release thru the 'exhaust' of the rocket/fuel container, and cause the rocket to move in the opposite direction.

Example:

Geo-stationary satellites have small rockets to keep them in place, and prevent or correct drifting, so that they can stay locked into their earth relay stations. Minute bursts of firing of these rockets on the satellites is done at an altitude of almost 36000Km above the surface of the earth, in what is near vacuum.