Snowmobiles won't be broken out until the park has been temporarily closed first and a lot of snow has accumulated on the roads.

Here's the webcams. You'll see there's no snow in the park yet: https://www.nps.gov/yell/learn/photosmultimedia/webcams.htm.

Like was mentioned above, it's possible that some services may not be available if there's a lot of snow. That said, I'd imagine the the lodges, bathrooms, etc. will still be open and staffed. If you're uncertain, I'd just ask the ranger at the entrance when you enter the park.

Keep in mind that while the entire park is at a pretty high elevation, some portions are higher than others so rain/sleet in one part of the park may be snow in another. Speaking from experience, if it's a cold rain (or snowing lightly) in West Yellowstone, you may have snow up at Canyon Village (Canyon is 1,000+ feet higher than West Yellowstone, I believe). We went last September and were getting rain at Madison Junction and in West Yellowstone, but as we hit Norris Junction it started to sleet and by the time we were 10 or so miles outside of Canyon it was a blizzard. When we got back to West Yellowstone it was rain again.

If the forecast is for rain or snow I'd just be sure to plan accordingly (warm layers, rain jackets, umbrellas, etc.). Maybe consider avoiding Canyon Village if you want to avoid snowy roads / heavy snow if things are already looking bad at the park entrance.