Five campgrounds in Yellowstone take reservations; seven do not. Check out US National Park Service - Camping in Yellowstone
for specifics. Bear in mind even the campgrounds that are first come, first served will probably fill fairly early in the day. As Bob said, even though it's September, the park is still busy; so you should plan your arrival accordingly.
We were there in late July/early August and got a site in Grand Teton NP, which is also beautiful and on the south end of Yellowstone. (We were in Lizard Creek cg, which is the cg furthest north in Grand Teton NP. It was okay, however the driveways in that cg weren't really designed for RVs. Since the Casita
is short, we made it work, but we were very close to our pickup, which had to be parked next to the trailer - instead of in front of it. Otherwise the campsite was lovely.)
One downside to camping at Grand Teton when what you want to do is tour Y'stone: you drive the same 40 miles twice a day to get into the park. It's a beautiful drive, but after a couple of passes through there we started to wish we were seeing other parts of the park we were not going to have time for.
As Tim said, the roads inside the park are good; however, the road between Norris and Mammoth Hot Springs is under construction (I understand the pavement melted from the forces of nature at work in the area), and I just noticed on the NPS website that the road between West Thumb and Old Faithful is closed. This might make staying in Grand Teton even less attractive - although if you have the time you should definitely go to Grand Teton and check it out, too.
HAVE A GREAT TRIP!!