There are other areas to cut wood. They don't list all of them. The local forestry office may be able to provide closer locations. For example, there is a designated area just a few miles from me but it is not listed on any of the forestry websites. I listed that page to show the permitting process, restrictions and costs. As toomuchtodo pointed out, it may even be free in their case.

There's a good chance that few of those areas are on the reservation, which often were picked to be the worst land in the region. So many still may not live within a convenient distance of an area for wood.

Plus, even if they do live near a forest, I'm not sure if the Forest Service would have jurisdiction. It's possible the tribe has the rights, or the BIA handles it.