The park is open year round, but some of the roads are closed at higher elevations. I know Cades Cove campground and Smokemont are open year round and may be reserved online. I don't know that reservations are necessary this time of year, but sometimes I check the reservations site before we go just to see how busy they are. :-) Neither of those have campgrounds hookups. Actually, they don't have showers either.
Winter is my favorite time to visit Cades Cove because the traffic is much more tolerable. You just have to watch the weather a bit. It can change drastically in a few days/hours. (just turned my air conditioner off and the heat on this morning. Will probably have the air back on this afternoon.)
Elkmont and most of the other campgrounds are closed for the season.