The nearest campground to the Banner Elk area resorts is Grandfather Campground. We have stayed there several times when skiing at Sugar Mountain. They are open year-round:
Grandfather RV Resort - Camping
We like it, it is relatively convenient, clean and prices are reasonable (though we have not been there over the Holidays where they seem to charge $5 /day extra). We usually stay for a week, they have a discount for weekly stays.
They also have WiFi for free, though we sometimes had trouble with consistent signal.