I've driven Sylvan Pass a dozen times or so (the road between Cody and Fishing Bridge), and you'll see snow & ice up there in July and August.
October 10th is when the majority of facilities are shut down, and some start in mid-September. It might be a risk, but it's worth visiting the park when the tourists are gone.
Just be observant of the weather forecasts. It's an awfully long drive back to Cody if you happen to get snowed out, and that entrance isn't open year round.
https://www.nps.gov/yell/planyourvisit/hours.htm