Turkey season opens here Saturday, or I would entertain this. It does not open in SEOK for 2 more weeks, however, so you should be good.
There is a backpacker's camp at Winding Stair. Trailhead parking is about 100 yards away, which makes this de facto walk in camping.
Winding Stair is an exposed, mountain top site. A more protected option is Billy Creek. Both sites are proximate to trails.