Camping in Yellow Stone in October

Yellowstone may be closed then because of snow.

Yellowstone is at 7000 ft. +
Opening dates here:
https://www.nps.gov/yell/planyourvisit/operating-dates.htm

Have you looked at this website? https://www.nps.gov/yell/planyourvisit/campgrounds.htm
By the way, it's spelled Yellowstone. Perhaps that was causing some of your difficulty? In the US a mobile home is frequently referred to as an RV, short for recreational vehicle. That may have also caused some internet search confusion. Also be aware that there are mobile home size limitations on some of the roads within the park.

Edit to add: the park is open year round but all potable water is turned off at the end of September.

Hi and welcome to the forum!
To add to the helpful info above, if you DO manage to find a campground spot in the park that's still open for the season, entry tickets are not needed. You buy an entry PASS, good for 7 days, and you can do that at an entrance station. Visitors entering through the South Entrance need to purchase an additional pass for Grand Teton National Park as they pass through it en route.

https://www.nps.gov/yell/planyourvisit/fees.htm

You could also look at campgrounds outside the park, Here's a list provided by the US Forest Service:
https://www.fs.usda.gov/detail/custergallatin/recreation/?cid=stelprdb5138825

Yes, you could see some snow.

If you planned to travel from Idaho east to Jackson and north to the park, let us know? There's a big road issue out that way but no need to mention it if you're not headed that way.

The recent road slide in Teton Pass will stop some access to Yellowstone and Teton Park from the SW of Wyoming for an unknown length of time. I would check to see if they have RV camping outside of the Park. If you are coming from Belgium, check other National and State Parks that are fully open in October. There are several that feature Autumn Leaves then and are not so hard to get to that time of year.

The road closure does not stop access to the park. That road is on the south side of Tetons heading from Tetons into Idaho.

There is only one campground in the park with electrical hookups which I would want in Yellowstone in October. fishing bridge on Yellowstone Lake. Have you looked at this website? https://secure.yellowstonenationalparklodges.com/booking/lodging-flex-search?destination=ALL-CAMPGROUNDS&adults=1&children=0&infants=0&animals=0&dateFrom=10-01-2024&nights=1. Looks to me like fishing bridge has availability the first half of October and Madison has availability all month. You can get by without electricity as long as you have a good propane heater.

We've camped in Yellowstone many times in our RV. Only been once in October and for that trip we did not camp, but it was very nice to not deal with summer crowds.

...heading from Tetons into Idaho.

Or vice versa. We don't know which direction they're coming from thus the head's up. That said, looks like a detour is already in the plans for the near future so that part can wait until they know whether they'll do Yellowstone or not, and from where.