Technically Arizona is a marketplace sales taxing states. Meaning, if you purchase anything online from a retailer, that retailer is supposed to collect tax and remit that tax to the state.
If you do purchase something, and get charged tax from the front end, you should not be paying taxes again once delivered. Not sure on if AZ allows a tax on services, like FFL rendering, but I'd question that if someone tried to charge me a tax on that FFL paperwork.
Believe it's up to you to file taxes for online purchases with the state if you buy from a dealer that is too small to collect taxes.
Where things get nebulous, is when you buy a used gun from someone out of state and ship it to you. This happened to me in Michigan when I used a FFL, who was also a LEO, who tried to collect tax from me on a gun I traded another couple of guns for. Was from out of state- I sent my two pistols to him, he shipped me his. The FFL wanted to use bluebook value for it.
It got contentious on both sides since he didn't let me know he was going to be pulling that BS move in the first place.
I've since gotten into the habit of asking a FFL I'm going to use if they are going to try and collect taxes on the service. If they say yes, I move on to someone else.