My wife and I both have carry permits and we are moving from Hennepin County to Wright County. How do we update the address on our carry permits?

MN Law requires that all people with that problem must take my "I have moved" Permit Class.

Or, you could instead follow this provision of the law....

Subd. 7a. Change of address; loss or destruction of
permit. (a) Within 30 days after changing permanent address,
or within 30 days of having lost or destroyed the permit card,
the permit holder must notify the issuing sheriff of the change,
loss, or destruction. Failure to provide notification as
required by this subdivision is a petty misdemeanor. The fine
for a first offense must not exceed $25. Notwithstanding
section 609.531, a firearm carried in violation of this
paragraph is not subject to forfeiture.

(b) After notice is given under paragraph (a), a permit
holder may obtain a replacement permit card by paying $10 to the
sheriff. The request for a replacement permit card must be made
on an official, standardized application adopted for this
purpose under section 624.7151, and, except in the case of an
address change, must include a notarized statement that the
permit card has been lost or destroyed.

As Phorvick pointed out in his citation, all this is done at the Issuing Sheriff.

Andrew correctly though perhaps a bit briefly linked to a good discussion on possible hiccups with Henn Co notices.