How are thermal (fire) barriers installed over foam insulation in a crawlspace?
Do I need to install the barrier in a crawlspace?


Yes you need to install the barrier regardless of where it is.


  What would I use as a barrier since I don't have studs to hang wall board on?
  Is there an alternative that I am missing?


You don't need studs to install drywall. You could install furring strips and then attach the drywall to this. You can even get XPS foam with channels for the furring strips cut out so that you don't add any extra depth to the wall.