J. E. Lighter, Historical Dictionary of American Slang, discusses several usages and origins, the first about the cartoon character introduced by Segar in "Thimble Theater," March 3, 1936 (Eugene the 'jeep').

'Jeep' is frequently used to designate a justice of the peace.

In an article in the Saturday Evening Post, July 16, 1938, 'jeep' was applied to an inexperienced person, and in the military was applied to a recruit or basic trainee.

In miltary usage, the term was applied to various small vehicles produced by Wilys-Overland. Research has shown that its origin as an alteration of GP for general purpose is unfounded. It was applied to the bantam vehicles used in the military (Infantry Journal), produced by Ford and Bantam.

In 1940, Q10 (and 1941, Washington Daily News), it was reported that 'jeep' had become a pet name for the "1/4 4x4" vehicle.

In 1942, 'jeep' was used for a small observation plane.
In 1942, it was reported in "Yank" that the 'jeep' was aplied to all small miltary objects.

In 1940, 'jeep' was applied to the Northrup N-1M plane.

The comments by GLoux about use in WW1 are interesting; I would like to see a full reference to the book. This origin is not mentioned in Lighter.