In the books, I can't remember what year it was, but Sauron sent orcs and other creatures to Moria, and then a couple hundred years after, Azog declared himself king of Moria. I think it's possible Sauron may have chosen an orc chieftain or champion to rule Moria. Is there any evidence to support or refute this?
I don't think there's any evidence that specifically support or refutes this. In the books, Azog was killed in 2799 of the Third Age, and Sauron had been relatively active (for a disembodied spirit of malevolence, at least) in Dol Guldur for a while by then.
I can't say whether or not Sauron personally put Azog in charge (That seems like the kind of minutiae that a god wouldn't get involved with), but Azog was almost certainly working for Sauron at this time. It seems extremely likely that Azog's killing of Thror (which kicked off the Battle of Azanulbizar in the books, rather than taking place during it as the movie shows), was motivated by Sauron's desire to possess the ring of power that he would later acquire from Thrain, Thror's son and Thorin's father.