Sheryl and Donna has been doing this for many many years i trust them when it come to finger printing.
Prints are not required to be taken by "a license and bonded official finger print services".
https://www.azdps.gov/services/public/cwp (This same site offers the application form)
"Note: You must utilize the "Applicant" fingerprint cards provided by the Department.
Some police agencies provide fingerprinting services for a small fee (DPS does not). Call your local police agencies for inquiries. Regardless of who completes the fingerprinting process for you, the prints must be clear, distinct and classifiable. Do not place tape over the top of the prints!"
The "STANDARD' print card is the FBI BLUE FONT FD-258
Also be sure to check the "written agreement" of the other state if that state has a written agreement with AZ.
There are some states that will only accept the AZ concealed firearm permit so long as the permit holder is a resident of AZ, a resident of the reciprocating/recognizing state or a plethora of other rules.
Be sure to contact the respective authority within that state.
Other states may be, if you'll pardon the expression, 'liberal" with their reciprocating/recognizing of other state's agreement, written or not.