Would an 89 accord dx carb. have a thermostat behind the glove box?

 

Yes, it wraps around the Evap case and sticks into the case to shut off the A/C if the temp gets close to freezing up the Evap.

 

would it keep the A/C system from working entirely if it was busted?

 

yes it would break the electrical path to the compressor clutch.
You might consider a schematic if you are trying to diagnose the system, those thermostats rarely fail. There are a lot more common problems, does the system have freon in it? have you check the rear window defrost fuse? That fuse is shared with the A/C clutch coil.