You find a seller, make the deal, fill out the Form 4, get it approved by your local chief law enforcement officer (CLEO), and send the feds the signed form and a check for $200. If your record is clean, no problem. M16s and selective fire AR-15s are a bit hard to come by and expensive, but are available.
AFAIK, AZ is a Class III friendly state, but some CLEOs refuse to sign the Form 4 on general principles. I suggest finding a local Class III dealer who can help you. However, on a purely intrastate (within a state) transaction, a dealer is not needed unless state or local law requires it. An individual can submit the Form 4.
Jim