No, you can't power the transformer from 2 sources.
A relay probably won't be any cheaper than a transformer and will be more complicated to wire.
If you want to go that way anyway, you would just get a relay with a 120v coil and run the coil wires to the hot and neutral supply going to your fan motor.
Then you would run a wire from the transformer to the common terminal on the relay, run a wire from the normally open contact on the relay to the damper. Then run a wire from the other side of the damper to the other side of the transformer.