You definitely won't be the only person getting around on a scooter. They are pretty prevalent.
The big issues I see are the crowds and the hills. Old Time Christmas is actually SDC's main season, and some times it gets so packed out that they close the gates early and don't let anyone else in. Inside the park it can get difficult in times to walk through, let alone ride through on a scooter. You just have to go slow and be patient.
The park is built on a ridge-line, and the West side of the park is the most hilly. Depending on the brand of your scooter, you may not be able to get up the hill on that side. The ADA-route follows the East side of the park. To get to the West side you either have to go all the way around by Fireman's Landing, or jog back past the train station to get to the circular ramp in the center of the park. It's a bit of a maze, so familiarize yourself with the park map.
The main "fat friendly" ride will be the train. I'm sure you can probably get on Flooded Mine as well, but you will have to go around to the back exit and send someone in to let them know you are coming. It's not setup well for ADA access.
Bathrooms are fairly prevalent, and they do have handicapped stalls. Again though, it will be very busy and some of the restrooms are hard to get around in a scooter even if they aren't packed. I think the ones near Crossroads Pizza and the ones at the back of Fireman's Landing are probably the easiest?
All-in-all, don't sweat it. SDC is magical at Christmas, and there's plenty to see and enjoy with your family.