We're from western NY and we've done Yellowstone twice using two weeks of vacation each time. In each of those trips we visited Badlands, Rushmore, Devils Tower, Yellowstone, Craters of the Moon and Grand Teton. It's 1900 miles to YNP from here so if you condense it to maybe just Yellowstone and Grand Teton, it can be done. We have dedicated "travel days"(about 700 miles a day)--a necessity for a trip like this. Do not listen to people who say it can't/shouldn't be done. If you're ok with the driving, do it. If you've never been there, it will be the trip of a lifetime, highly recommended!!
Sights in YNP; Old Faithful(of course), Lower Falls, Artist Point, Travertine Terraces in Mammoth Hot springs, Grand Prismatic Spring.
You might have luck in Grizzly RV Park in West Yellowstone but I'd call ASAP, the CGs in YNP will be a challenge to book this late. Colter Bay cg(not the one with hookups) in Grand Teton usually has spots open. Good luck!