Yellowstone is a very big park. You likely will stay in 2-3 locations for that park. I've experienced temperatures as low as -45 F in Yellowstone, so you need to be safe and prepared for weather with emergency supplies. There is no cell phone coverage in most areas. I'd plan a trip of at least a full week so you can adjust to weather. This year has had minimal snowfall, and when that happens wildlife continues to graze in higher elevation areas rather than wintering in the valleys. Other years you may have 5 feet of snow on the ground, and it makes seeing wildlife much easier since animals concentrate in lower elevations and can be seen more easily against snowy landscapes.
Gardiner, Montana on the northern entrance to the park is close to Mammoth Hot Springs and the road to Cooke City which passes through Lamar Valley. The road from Gardiner to Cooke City is maintained for cars throughout the year, and you can normally drive on it in a normal car even in years with a lot of snow. Still, I would recommend an SUV with 4 wheel drive. Lamar Valley is normally a very good location for wildlife and landscapes and with good activity justifies several visits. Gardiner to Cooke City and the return is a full day activity. The nearest airport for Gardiner is Bozeman MT.
West Yellowstone is the normal location for commercial snow coaches as well as entry to the geyser areas such as Old Faithful. Transportation is by snow coach or snow mobile only. These areas are known for wildlife - especially bison around the geyser areas. You'll also possible see birds and elk along the Yellowstone River. You need to arrange commercial transportation for this area and a snow coach is normally the preferred choice. Snow coaches can be overfilled by tour operators so be sure to pick a vendor or guide that caters to photographers. West Yellowstone is a 3-4 hour drive from Gardiner all outside the park.
The third and final area is not in Yellowstone at all - it's to the south in Grand Teton National Park. Wildlife is a bit more accessible at Grand Teton. You can normally use an SUV for travel, but roads may close with heavy snow. There are lodges as well as commercial photo leaders in the area, but you can visit Grand Teton by yourself. Jackson WY is a ski town and is open and busy throughout the winter. There is a small airport just outside Jackson with car rental.
NANPA (North American Nature Photography Association) has been hosting trips for members at discounted rates led by NANPA members who live in the area. It's a good option for a three day trip, then do the rest on your own. Visit NANPA.org for details. I think they are just starting a sold out trip for 2022.
http://www.nanpa.org/events/regionals/yellowstone-snowcoach-2022/
All three of the NANPA leaders are excellent. You might also look at Trent Sizemore - a young pro and NANPA member living in the area.
