Air vent door keeps opening and closing after unit stops.

Exhaust Fan Motor

If the exhaust fan does not run the fan motor might be bad. Remove the fan motor from the microwave oven and see if the blades can be turned by hand. They should turn very freely, if they're stiff the fan motor will need to be replaced. The fan motor can also burn out. To test this use an Ohm meter. The motor should have continuity. If it doesn't, replace it.

Thermostat

If the microwave exhaust fan is not working the thermostat might be defective. The thermostat senses the air temperature and turns on the fan as needed, if it's defective it won't turn on the fan.

Main Control Board

If the microwave exhaust fan is not working check the exhaust fan motor and wiring. If they are good the main control board might be defective.