We've driven it with our 40'. No problem except for a bison roundup by wranglers which was awesome to watch... right in the park. They were trying to get them all from one side of the road to another and they came right by our MH.
Here's an interesting read on all the road entrances:
Road to Yellowstone
Here's the park's web site with lots of good information on things to do, special alerts, etc. Read it thoroughly so you'll have no surprises. It also lists the 2018 road construction schedule. They can only do work during the busy summer season so there are always delays somewhere in the park.
https://www.nps.gov/yell/planyourvisit/parkroads.htm