At the Mt. Barcroft gate trailhead (12,000') there might be enough space to pitch a few small tents, and there are couple of fire rings. No reservations are needed for camping but a campfire permit is required and available from USFS office in Bishop. No toilets and no water.
Because of the heavy snow-pack this year, there could be road damage during the Spring runoff. Check with the USFS White Mountain ranger station in Bishop before travelling up there.
