Well, the government is shut down, but you'd never know it out on the prairies and ridgelines of the great outdoors. It's hard to decipher exactly what is shut down from the Forest Service's cryptic pronouncement here, but a quick lap around the internet seems to indicate that most of the Forest Service websites are still up. Try the main Forest Service page here and click on the forest near you. I'll run through a quick and dirty method of finding camping opportunities outside the shut-down campgrounds in the national parks.
I don't know for sure about the status of the National Forest campgrounds themselves – if it's a concessionaire running it, it's probably still open. There are exceptions – Kirk Creek and Plaskett Creek in Los Padres National Forest in Big Sur are closed. Some of the non-concession national forest campgrounds may also be open – unless they're barricaded, fill out the envelope and drop the money in the box and you've got just as much right to be there as anyone.
The best bet is to look at dispersed camping, as I mentioned in my earlier post on the topic. Go to the maps page of the National Forest website, click on your forest, and look for Motor Vehicle Use Maps (MVUM). These are designed for off-road vehicle users, may the gods curse their worthless hides, but the collateral benefit of these maps are the dots which designate dispersed camping opportunities. Any road marked with these dots is fair game for boondocking.
I wish the ranger stations were open – I like to check in and see what roads are open, how crowded they are, bear activity level, snow warnings, etc., but if you've driven all the way out to a National Park to find the road blocked, check out nearby national forests. For instance, Gallatin NF is north of Yellowstone. Here are links to Motor Vehicle Use Maps for the entire forest, and there are dots all over showing the forest roads where dispersed camping is allowed. Stay low in altitude and close to the paved roads and you will do OK. It's better than turning around and driving back home.