If your grandfather has the WWII marker then the local VA office might have information about his service especially if he used any G.I. Bill benefits or other VA benefits later in life. Also contact the local VFW (Veterans of Foreign Wars) chapter to see if he was a member.
FYI: If the Veterans Office says they can't find any records with your grandfather's birthdate as you know it, month/day/year, and both the day # are 12 and under then resubmit it substituting the day for the month and month for the day. Example: If his birthday was 11/07/1928-Nov. 7, 1928 then use the date 07/11/1928-July 11,1928. According to my dad, a retired Navy man, the Veterans Adminstration used Air Force personnel for computer data entry and they used the day/month/year format for thier records and the Navy always used the month/day/year. All his computer records with the VA now have the "wrong" birthdate as he refused to, "jump though their hoops to correct their mistake". He also claimed to have talked to others using Veterans Services with the same problem.