You only need an archery permit to hunt during an archery-only season. There are no "archery-only" turkey hunts, just general and controlled hunts.
To hunt turkeys with a bow during a general season, you need 2 items: a turkey tag and a valid nonresident license (3 license options: 1. Hunting, 2. Small Game Hunting, or 3. Three-Day Nonresident Small Game Hunting).