www.recreation.gov
As I recall, the White Mountain National Forest has about 23 different campgrounds spread across its mountain forest that cost something like twenty dollars/night for a camp site. Because it is federal property, under the direction of the US Forest Service, Dept of Agriculture, it does not have the NH-9% rental tax similar to state-NH or private-NH campgrounds.
It's decision to open or close is probably its own US Forest Service decision made separate from the NH government decision and it's a little difficult to determine, but it appears that all Forest Service campgrounds are closed for now, till further notice.