I underallocated the amount of disk needed for my local bes relay server. If I allocate an additional volume is there any way to configure the relay to use the new volume for the files located in C:\Program Files\BigFix Enterprise\BES Relay\wwwroot\bfmirror\bfsites?
Yes. You can muck around with some of the registry keys that point to the location of the file stores, but that is error prone and will likely be reverted if you upgrade.
I recommend that you uninstall and re-install the BES Relay on the new drive. You can do it manually or thought the Tasks.