Few things to keep in mind. In May I believe only the North entrance is open and I'm pretty sure many of the roads will be closed. I *think* they keep the main loop open but I'm not sure the Washburn pass will be open. Anything high elevation will be completely covered in snow and ice.
That being said here are some options:
If you are outside the part, I believe you said you were in ID, I didn't look up the town but you'll probably be closest to West Yellowstone. There should be places to camp there.
If you are northish you'll want to go to Gardner, MT. There are plenty of campground/hotels there too.
If you want to be IN the park, there are only a few officially registered campgrounds. Just check the national park website for which ones are open during May. I did a quick check here are your options(Campgrounds - Yellowstone National Park (U.S. National Park Service))
-Mammoth(this is by the hotsprings)
-Madison(by West Yellowstone entrance. Not much "elevation" hiking here, pretty spot though. Dense woods)
-Fishing Bridge(the biggest campground, right on yellowstone lake. Nice central point honestly and a badass diner right there!)
-Norris(decent location, lots of geyser basins, not the best for elevation hiking but not a bad hopping off point)
-Looks like the rest don't open until after you will be gone
Personally, from that pick list I'd recommend FishingBridge, Mammoth Hot springs is worthwhile to see, it's amazing to see nature like that. But it's easily the most tourist spot in Yellowstone. Fishing Bridge is larger, more "modernized" and not very "naturey" but it has a great location for getting to many spots in the park, plus you can stop at all the main attractions on the way there, knocking out the tourist crap in one quick day.
As far as hiking elevation, again keep in mind anything up high will be buried in snow and ice. I havn't kept track of the snow fall this year but 2m+ is not at all uncommon in this area. I'm not here to lecture you so let's just say you know what you are doing and are prepared. As you probably know with Yellowstone being a national park they prefer everyone camp at designated backcountry sites. If you've been out at all you know this and at what level they "enforce" it
Here are some decent elevation hikes:
-Touristy but the main mountain is Washburn, if you just want a quick little hike that can offer a good vista(center of the park kinda) it's a neat one.
-Bliss Pass Trail is nice, plus it's a got a beautiful approach and termination hike through some nice valley's. This is in the NE of the park
-Specimen Ridge Trail, basically you ridge walk for about 14 miles. It's beautiful and if you are lucky and it's nice out you can see some amazing things. Also in the NE but south of Lamar Valley, which has tons of wolf/bear activity. The very east end of this trail where I camped out south in the valley of the lamar valley is the only time in my life I encountered a grizzly in the pure wild(not the yellowstone style where they are alone the road and everyone stops to take pictures)
-In the Nw of the park look into Sportsman Lake Trail. The trail itself doesn't climb mountains but you hike into this cool valley and there are all sorts of side hike opportunities that go up into awesome elevated lakes. Fawn Pass Trail and BigHorn Trail run parallel to it and I'm sure they rock too, just never done those ones.
Anyway that's my personal "elevation" experience in yellowstone. Honestly though, I don't hike in yellowstone to conquer mountains, I do it for the wildlife and crazy diverse eco-system. I've been on 7+ day hikes there that started in deserts, went through valleys, into woods, over a pass and then stopped at yellowstone lake. It's just amazing how it changes there due to the geothermal stuff.