My grandfather fought in the Army in WWI. How can I obtain a copy of his military and combat records without a SSN or service number?

It sort of depends on what unit he was in, if he were in one of the National Guard units federalized for The Great War, your state adjutant generals office should have the records, if RA, I have no clue but would start with the National Archives.

First, its NOT impossible to get his SSN. I'm assuming he has passed away. Since WWI was so long ago. If so, there is a Social Security Death Index available on line. You put in his name and birth date and some other info like what state he died in, etc, and you can narrow it down and get his SSAN number. Otherwise, I'd suggest going to the local Veterans Affairs person, and asking for help. The record will be at St. Louis. Then, as a former personnel specialist, I'm telling you it might be a crap shoot. Because St. Louis has had several fires, and some records were destroyed. Also, let me know if you want a little bit of a hand. I have an account with Ancestry.com and I would be happy to help you find his SSAN. I mean, I paid for the subscription, so I might as well use it to help someone.

There was a big fire at a records building in 1973. Many records were lost forever. That said, my mom got my father's honorable discharge from the Marines in 1919 and from the US Army in 1943. Alas, she is no longer around to ask how that worked.

National archives has a great web site and will help you find records. Give it a try.

Someone mentioned the ST Louis fire where many Military Records were destroyed by fire Here is a link that describes what was lost. http://www.archives.gov/st-louis/military-personnel/fire-1973.html