You're going to want to make sure you can use your A/C. It can still be pretty hot in mid-September. The campgrounds run by the Park Service don't have any hookups. Cottonwoods has a no generator policy. The only place you can get full hooks are at the Rio Grande Village in the commercial campground. They have about 30 spaces on a paved parking lot. You may want to make reservations.
Alternatively you can stay at one of the many commercial campgrounds in Study Butte just outside the Park on 118. You'll probably want a toad to visit the Park. A 4-wheel drive is best because there are a lot of roads you can take then to follow the river or visit old town and home sites.