Yellowstone in late September can be fairly cold, with chance of snow too. Most of the lodges start shutting down in mid-September, and some of the campgrounds too. I've been there a couple of times in the fall to do photography and it's really nice. Animals are out and about getting ready for the winter, elk are bugling, and there are way fewer people about. However, I've seen it get down to 15 degrees at night with snow in the first week of September, and be 80 degrees on another trip in the middle of September (I had to buy a t-shirt because it was so hot!).
While I'd normally say "go by bike," for this trip I think a car will be much better, given the short daylight hours, weather variability, etc.