If you shoot light out the backside of your craft then indeed you would move towards the front. But then you'd use up some energy in order to go because the light that leaves would carry energy as well as momentum. The energy of your craft would decrease and thus the mass of your craft would go down. You'd arrive at your destination with a less massive craft than you departed with. But some people might want to move without having to lose mass.
A quantum vacuum plasma thruster (or Q-thruster) is a name given for a variety of devices that claimed to be reactionless drives. A reactionless drive claims to emit no matter or radiation. It could arrive just as massive as it departed, without having to pick up fuel on the way there.
For example, if you have a cavity inside your craft and if you emited light from the top of the inside of the cavity towards the bottom of the inside of the cavity, then at first the top would go up but then when your light hit the bottom the bottom would start to go down. Since the light is always inside the cavity you could claim that no light or energy or momentum gets outside, so it's reactionless.
The drive you imagine where light comes out the bottom, is not a reactionless drive because you propose to emit radiation. And most people are not excited by proposed reactionless drives. So does it work? In some sense you could sit on one end of a box in space with a bunch of super balls and start throwing them to the opposite wall and the outside of the box will start moving. The center of mass doesn't move but since there used to be a pile of superballs on one side the outside can move one way as the center of mass of the superballs moves the other way. So sure, you could expect some very small motion of the container if the energy inside moves itself about. But nothing exciting. What do people see? Here's an example of a report.
An article published by Shawyer in Acta Astronautica summarises the existing tests on the EmDrive. Of seven tests, four produced a measured force in the intended direction and three produced thrust in the opposite direction. Furthermore, in one of the tests, thrust could be produced in either direction by varying the spring constants in the measuring apparatus
https://en.m.wikipedia.org/wiki/RF_resonant_cavity_thruster
Almost anyone that looks at such results thinks the results are noise. And the claims on the face of it look like obvious violations of conservation of momentum. So almost no one pays any attention at all (except in the sense of concern about science outreach and how public perception of science might be affected by phantastical claims).
And keep in mind that terms like "quantum vacuum plasma" are not standard scientific terms.
