From our experience last year, Cody rolls up the sidewalks on 9/30. We got there a day or two after that, outrunning a blizzard from Gros Ventre Campground in Tetons and found one CG still in the process of shutting down. They let us stay overnight. There was 8 inches on snow on the roads in the park and the snowplows were active.
It looked like the State Park CG may have still been open as we drove past, the one to the west of Cody. Not sure tho.
We stayed the last four days of the season in Gros Ventre, they were closing down as we left.