If you're intending to stay in the actual campground, you're supposed to check in with Parks & Recreation and let them know you'll be doing it, but it is open.
Having said that, there is no particular advantage to being in the campground proper, as opposed to an LNT site in the surrounding state forest, since none of the facilities will be open or working -- and if you're at large in the state forest there is no need to get permission in advance. If it were me, and my purpose were to establish a hiking / skiing basecamp, I'd go up past the group campsites (a bit above the regular tentsites and leantos) and head into the woods just south of there.
The road is very pulkable -- to give you a sense of its gradient, it's the sort of place where you can ski turns down it if the snow is fast and compact, but in powder you may need to run a straight line or even kick and/or pole a bit.
Have fun!