http://www.atf.gov/files/publications/download/p/atf-p-5300-4.pdf
Page 7 of the above document:
Federal Law 18 USC 921 (b):
(b) For the purposes of this chapter, a
member of the Armed Forces on active
duty is a resident of the State in which his
permanent duty station is located.
Page 39 of the above document:
27 CFR (Code of Federal Regulations) 478.11:
State of residence. The State in which
an individual resides. An individual resides
in a State if he or she is present in a State
with the intention of making a home in that
State. If an individual is on active duty as a
member of the Armed Forces, the individual's
State of residence is the State in
which his or her permanent duty station is
located.
Pages 135-136 of the above document:
ATF Rule 2001-5:
Held further, a purchaser who is a
member of the Armed Forces on active
duty is a resident of the State in
which his or her permanent duty station
is located, and may satisfy the
identification document requirement
by presenting his or her military identification
card along with official orders
showing that his or her permanent
duty station is located within the State
where the licensed premises are located.
All of the above applies ONLY TO THE MILITARY MEMBER and not the spouse or other dependents. In order to purchase a firearm other than a rifle or shotgun from an FFL in the new state, they must obtain the "standard" proof of residency such as an in state driver's license or state ID card - or present some other government issued document that proves their new state of residency. Spouses and dependents are not issued military change of station orders. I would recommend you read the entire section on pages 135-136 of the Federal Firearms Regulations Reference Guide as it contains complete information regarding military members state(s) of residence.
In Minnesota, the permit to purchase a pistol or resident CCL would be an adequate government issued document for the spouse to prove residency to an FFL and they could use any other government photo ID to prove identity.
It won't be long before someone comes along and confuses "home of record" state with state of residence. I have had a Wyoming driver's license for 30 years now. My military "home of record" state is Wyoming. For the purposes of firearms transactions, I have NOT been a resident of Wyoming for 20+ years. Where you are registered to vote, where you pay taxes, and where you maintain a driver's license may indicate what your state of residence is - but it does not establish your actual state of residence for firearms transactions - actual state of residence for firearms transactions is where you sleep at night, and for military members where you are ordered to go to work every day.