According to my version of the map, there are several locations with indoor accomodations. Some of them are listed as "cabins." Who knows what that includes?
One issue I see is that, going counter clockwise, there does not appear to be any indoor accomodations between Paint Bank and the USFS Middle Mountain cabins. That's a distance of nearly 105 miles. To find something sooner, you would have to take either the Dumore or Bartow cutoff.
(The place in Paint Back has an old caboose you can rent. Looks like a popular place where a reservations would probably be a good idea: http://www.depotlodge.com/reservations.php
You might cosnider sprining for the map (it's available for download) and contacting the places for accomodation details. That would also allow you to check the spacing of other indoor options. Looking into reservations for places along the Greenbrier Trail miight also be a good idea. None fo the places sound like large establishments.