There is no constitutional provision for the governor to make appointments to the House. The seat will remain vacant until a special election can be held.
Arizona Constitution?
Arizona Statutes apply ALTHOUGH THIS IS PREMATURE:
16-222. Vacancy in the office of United States senator or representative
A. When a vacancy occurs in the office of United States senator or representative in Congress by reason of death or resignation, or from any other cause, the vacancy shall be filled at the next general election. At such election the person elected shall fill the unexpired term of the vacated office.
B. For a vacancy in the office of representative in Congress, if the next general election is not to be held within six months from the date of the occurrence of the vacancy, the governor shall call a special primary election and a special general election to fill the vacancy. The governor shall call the special primary election and establish its date within seventy-two hours after the office is officially declared vacant.
[then a lot of how-to clauses]