Unless they are experienced at using a
scooter (also called
ECV), I would not suggest trying it at Christmas/New Years. Even during slower times, it takes some getting used to to drive one in a busy area.
We have been to WDW a number of times during that time period (we had not choice, DH wanted to go to Orlando to see the bowl games on New Years). Our last trip was this year. It was busy enought that several of the parks closed to anyone coming into the parks on at least 2 of the days (so it is a possibility to leave and not get back in). We were pushing a wheelchair thru the parks and found it to be a bit like Christmas shopping with a wheelchair; crowded, busy and took longer to get from place to place. The waits for anything were long - attractions, food, bathrooms, so you will need extra time for everything and can't plan on doing everything you might like to do. Having a wheelchair or ECV does not get you any special placement in lines. It was busy enough several times that you could not really choose where to go, you just follow the crowd (like a school of fish).
Because of my DD's needs, we can't get to the parks early, but that is a very good thing to do. Other members of our party have done that during busy times (Easter, on several past trips) and found they could see most of Fantasyland and several of the other high demand things (Splash and Thunder Mountain) with almost no wait. Since the parks are open late most nights, many people are up late and don't get going in the morning. If you do that, you can leave the park once it gets busy and go somewhere else - visiting the resorts to see their Christmas decorations.
If you have to go during that time, I think using TourGuide Mike (a Theme Park Board sponsor) would be a good idea. We have nver used him, but I have read that his service allows you to figure out the least busy places to be so that you can be there.