Elmo the Spy is a freelance agent.
This is an assumption based on very little data but it seems a sound one as the two escapades we follow him on certainly look like he is an operative of opportunity rather than under orders for some particular organization. In the first he is approached by a would-be overthrower of an Indian sub-continent region to help oust the sitting ruler. In the second, it is a get-rich-quick scheme involving extortion that occupies his attention. While these two cases take place in the mid-50s when the Cold War was becoming especially chilly, there is no indication that Elmo works for 'the other side'
Elmo is a natty-dressed man with a pencil-thin mustache in the first adventure and a shortish handlebar mustache in the second. Most importantly to the image of his being a no-good-nik, he favors a monocle on his left eye. He always sports a fedora which sort of kills the ominous vibe. He is clean-shaven with a long pointy nose and a chin that follows that trend.
We are presented these two missions through the good graces of a young pesky, annoying youngster with the name of Winky Dink. We do not know if Winky is a nickname for this little fellow with star-shaped pointy blond hair or his real first name - he does have a female cousin he calls Twinky.
To say that Dink is a buttinsky is to be kind. In the pages of the dossiers we are shows of him, he gets involved in all sorts of troubles and shenanigans totally unconnected with Elmo the Spy. This is mentioned only to show that Dink is always being a scamp somewhere and it is just the misfortune of Elmo that he runs into Dink.
Back on Elmo the Spy - this man is devious and creative and all-around seemingly a very resourceful and experience operative. Which makes one wonder why it is that someone like Winky Dink - whom statements show is already well known to Elmo and vice-versa - can cause him so much trouble!