Yellowstone - inside park campground recommd 30 ' trailer

Here's all the info you need on every CG inside the park. Opening dates, Fees, # of sites, Elevations, Special Features and Notes.
As far as "central location" goes, the road is a 60 mile loop. At any place on it, you're as far away as you can be from another point on the loop.
BTW, September is a great time to visit. The kids are back in school, so the crowds are smaller, and the leaves are starting to turn.