Closing dates for Yellowstone roads are Oct.9th for Dunraven Pass and the Beartooth Highway. On Oct 15th Fishing Bridge (the bridge and road, not the campground, it is already closed) will close for replacement. That means you will no longer be able to access the loop roads from Cody.
If I was me, I would absolutely make the trip if it was a one day drive and you were willing to accept the fact you might be stuck for a day or two should weather move in and you are comfortable dealing with below freeing temperatures and snow covered roadways. If it was a two or three day trek, and it was your vacation for the year I say it is way too late in the season to risk a trip. As others have mentioned, it isn't only the park that is closing down, it is all the businesses in the surrounding communities are closing as well. For some, it's goodbye till next year. For others, it's a breather before they have to scale up for Snowmobile season (West Yellowstone) or Ski Season (Jackson Hole).
I also would only consider full hookups. If the weather turns cold, dry camping will tax your RV heating and plumbing systems. It might be a long drive to find a propane refill in October. Your best bet is going to be West Yellowstone. There are a couple of parks that stay open into Mid October. But I think you will strike out if you want to stay into late October. Even the late closing parks are going to be closed up tight, for good reasons.
