That relay is commonly found in the glove box of F150s with the tow package. A relay like that, once placed in the appropriate fuse block, provides power to charge the battery in a trailer (like my horse trailer, for E-brake).

 

Most cars have several of the same part number relays that all do different things. One thing a person can do when troubleshooting a problem is to simply swap relays and see if the problem follows. As far as your a/c goes you should be able to start the engine and stick your head in the engine compartment while someone turns the a/c on and off. This should make the a/c clutch open and close which you should be able to hear. If you don't hear anything then I'd guess you should try a relay.