I know someone who was supposed to report to work on the last week of April, and arrived to find the campground for employees had been 'cleared' of snow, well the paved sections, but not the snow in between sites. He could look out his bunk window in the fifth wheel and see snow at eye level. So I am guessing well over 4 feet deep. He said it was a few more weeks before it melted away, and started to warm up.
Best to call a resort, and ask the question in a few weeks. West Yellowstone has a few full hookup resorts (in Idaho) that are just outside the park boundary. They are in a city large enough to be open all year long.
The snow level can vary a lot each year. This year might be light to really deep, you have to call a local to find out, but next month. As it might change in March. It is a fairly high elevation, so there is a lot of snow each winter. The Beartooth summit pass is 10,000+ elevation, so is sure to be snowed in in May. The roads might be clear, but probably 2' of snow on all the interesting fire roads, where the best dry camping areas are located.
Good luck,
Fred.