No problems going thru the park on any of their roads. If you are coming in the East entrance be sure to stop at Cody to see the Buffalo Bill Museum. We spent two days seeing it and stayed at the Buffalo Bill State Park outside of town.
If you are coming in thru the South entrance be sure to spend some time in Grand Teton NP. Stay at Gros Ventre Campground.
We have been volunteer camp hosts in Grant Village for years and regularly see large RV's in the park and passing thru.
I hope you have allowed lots of time as there's lots to see in the area.