AZ still issues permits and as previously mentioned they can be had by mail. Go to the AZ DPS website and either email ask for a package or call.
When you get it fill out the simple one page form and send it back with the two print cards, 60 bucks, and a copy of both sides of your Montana permit or alternate acceptable documentation as specified on the website.
They used to turn permits around in less than two weeks but these days it's taking a little longer due to demand.