I always stay at Fishing Bridge CG, for the convenience as much as the full hookups. This is at the SE of the figure 8 main road. One week is adequate to see the major attractions.
For how late in the year? Anytime after mid-August it can snow enough to close the roads. One must stay informed to the weather there.
If you can't get reservations for a week, try for 1 day or as much as they have available.
I have decided on a whim to visit Yellowstone
, called and got 1 day(night). Once there, I am first in line next morning to claim a cancellation, of which there are numerous, because there is no penalty for canceling reservations. I may have to move campsites once or twice but staying a week is rather easy without advance planning.