If a conviction has been vacated, does it need to be disclosed on an employment application and will it appear on a background check?

It is probably a good idea to disclose and explain that the conviction has been vacated and what that means.  In Arizona, when a conviction is vacated or "set aside," the record still exists.  Therefore, it may very well show up on a background check, but it should say that the conviction was vacated or set aside.