I finally got around to getting my AZ permit.
When I emailed a request for an application packet, they said everything is now online, print and mail. They didn't specify where to get the fingerprint cards, I ordered these for $4.99:
https://smile.amazon.com/gp/aw/d/B00...b_b_asin_image
And this for $6.98:
https://smile.amazon.com/gp/aw/d/B00...b_b_asin_image
Sent in a copy of my CA permit and a copy of my training cert from the class I took for my CA permit.
Received AZ permit in 9 days.