I think it will depend upon if it's actually the Park Service that handles the reservations, or if that's contracted out to a private concessionaire.

If it's the Park Service, there probably won't be anyone to respond until after the shutdown ends.

The good news, if it is the Park Service, everyone is having the same issue as you are and nothing will get booked. Also, there are lots of great campgrounds around the Moab area if Arches does fill up. The Campground at Canyonlands/Island in the Sky is one of my all time favorites, and it doesn't take reservations.