Audi Repair Questions? Ask a Mechanic for Answers ASAP
This flap is controlled by the temperature knob / dial etc and I think is controlled with a small motor for your car so the thing to do is first get access to the back of the control knob and measure the voltages is present there, then trace the wiring back into the dash from the control to locate the motor.
It’s worth first checking the fuses for the fan / control circuit
I'd recommend first removing the glovebox ( undo lower edge hinges from dash) to gain access to the dashboard inner and also allow you to shine a torch to the back of the central console so you can start tracing the wire back and see how things are put together