Once exhaust comes out, space is no longer a vacuum, at least not in a small area for a short time. More importantly, there is no physical connection between the exhaust and the rocket like there is between your foot and your ass. If you kick yourself in the ass, your foot will basically be pushing off of itself, try to drag itself along, and Conservation of Momentum will leave you stationary.
With a rocket, a small mass goes one way at high speed which causes a large mass to go in the opposite direction at relatively low speed. Both masses push off from each other. And this continues as long as the rocket keeps shoving mass out.
Some science fiction speculates “reactionless” drives that operate without ejecting reaction mass, but those are purely fictional as they violate the laws of physics as we currently know them. Somewhere in the middle are drives that eject less mass at higher velocity, allowing for comparable thrust with less fuel required and thus a smaller, lighter spacecraft, but many of those rely on things we currently lack, like fusion reactors, or antimatter