I posted this in another thread on the same topic:
Indy is an American patriot and he hates Nazis, he would have served his country in WW2 without question. His knowledge of history and languages would make him a perfect candidate for an intelligence officer.
However, Indiana Jones was a world-renowned archeologist. Heck, Chattar Lal recognized his name almost immediately. As such, I do not feel that he would have been a "behind the lines" sort of operative. As has been mentioned, he has killed so many Nazis by 1939 that his face must have been plastered all over Germany. The Gestapo may have even had people out looking to kill him. What Nazi is going to fall for it when Indy shows up in Germany undercover? My thought is that Indy would have been attached to Army Intelligence, but stationed Stateside or in England, analyzing information and decoding German communiques with Eaton and Musgrove. He might have occasionally been sent to recover artifacts whilst racing against the Nazis, ala ROTLA and TLC, but for actual behind-the-lines operations and commando work, Indy was too well-known, and frankly, too old.
I also agree with Lance that after the war was over, Indy would have returned to academia and archeology, not continued to "spy on the Reds." However, Indy as a post-war spy is at least a little more plausible, because he could have been stationed in Berlin, ostensibly as an archeologist or academic, while secretly gathering info on the Commies for the US govt.