We've been there in Nov & Jan but never Christmas so someone else will have to give you info on how popular it is at that time. I would recommend you reserve a site at Rio Grande Village campground - dry camping and very nice (or RV park, next door, if you need hookups. It's a straight line of RVs on blacktop).
Rio Grande Village has a small grocery for staples, a laundramat and showers. Also, expensive fuel. Fuel is also available at Panther Jct. as you're driving through and it might be a little cheaper. Rio Grande Village is a very pleasant area with some nice little hikes from there and it's a flat area for a little biking riding.
Big Bend is huge so you won't be riding your bikes to get to all the areas and they don't rent horses. However, if you drive your Class C to trailheads you'll be fine as most would accommodate you.
Have a great trip!