a quick online search takes me to the DMZ website; Out-of-State Replacement
Out-of-State or Military License Replacement
You can replace a lost
Arizona license while out of state using the online, telephone, or
mail-in options outlined above.
The MVD allows the same processes for military members deployed out of
state and their dependents, too. Simply apply for your military
replacement license online, by calling the MVD, or by mail—whichever
is most convenient for you.
If you apply online, you will need to choose the express delivery
option for an additional $16.95 fee (see “Apply Online" above). This
option lets you order delivery to an out-of-state address. Otherwise,
the duplicate will be sent to the address on file with the MVD.
The out-of-state address you provide for express delivery will NOT
appear on your duplicate license or ID, nor will it change the
permanent/resident address on file.
Remember, express service is not available for military delivery boxes
or P.O boxes (e.g., APO, FPO) and is only available in the U.S.
You can apply online at the Arizona Department of Transportation.
You do need an address they can send it to, however its likely to take up to 15 You may be able to get it delivered express which may be much faster.
How long will it take to get my Duplicate Driver License or Identification Card?
You will receive your Duplicate Driver License or Identification Card within 15 days.
If you request express delivery, an expedited delivery date will be provided based on the state and zip code of delivery and the time
the order is placed.
Given you are on a road trip, this may or may not work for your time frame. Give it a try and see what the expected date of delivery is. Otherwise your friend best avoid driving without a license.
She can of course, order a replacement to be sent to her new Oregon address (which she has done). We can't however advise you on the legality of driving without a license in this scenario. Obviously there are other ways to continue her journey without driving.