Check the Veterans' Administration website for specific help in making a request.
After sending the request, expect a response that says they don't have records because of the 1973 fire. If that happens, send another request!
My second request got to a second NARA researcher who DID find my father's military records. They were damaged in the fire, but portions survived in good enough shape to begin to piece together his record.
Be prepared to pay a small fortune for the copies. My Dad's record cost me $50, and that was almost 2 years ago.
All that said, it's still worth asking. Be sure to give them all the information you have about the veteran. If you don't know the actual dates of service, try to provide a reasonable estimated timeframe (and be sure to say it's an "estimate").