Several different good ways to get to YNP from I-80. In mid-May, some of the higher elevations may not yet be fully plowed, depending on the kind of winter we've had. Interstates are probably likeliest to be cleared, in which case you could take I-80 across Wyoming (watch out for wind and occasionally snowy conditions) and catch I-84 in Utah (less of a climb than I-80, as I recall) to I-15 north - then US 20 from Idaho Falls to West Yellowstone.
An excellent campground with hookups, just outside the park, is Grizzly RV in West Yellowstone. We usually use it. You will need reservations well ahead of time, and they may be taking them already for next May (not sure). You're not IN the park, true, but much less crowded conditions than in Fishing Bridge.