Sept. is a good time to visit Yellowstone. Elk are bugling, less people(normally) great fall colors.
Campgrounds should have vacancies,but make reservations if trying for Fishing bridge or Madison.
Long trip not to have your own toad.
Weather can be anything, come prepared. Don't forget your tire chains! Never used mine, but required to have them.
Fuel is available, however try to get it before getting into park as price goes up. Some park stations will be starting to close.
Construction in some parts of the park will be continueing. Traffic delays are possible not counting animals traffic delays.