I don't know anything about the Fort Wilderness Campground carriages. But the Wilderness Lodge "sleigh" was basically a carriage that had runners that made it appear like a sleigh. We went last Christmas and had a very nice time. The route they took us started in front of the Lodge and went around the side into the wooded area.
I've seen the carriages at Port Orleans Riverside and the carriage had large wheels, so I'd think that Fort Wilderness is probably similar.
The appearance of the sleigh wasn't a big deal for anyone in my family. I think it probably wouldn't have made a difference if it had been the carriage. If we were to do it again, the route that the carriage takes at Fort Wilderness vs. the route that it takes at the Wilderness Lodge would be a bigger factor for us.
Good luck!