The use of the word for the small carriers, 'jeep carriers,' also is included in the Lighter "Historical Dictionary of American Slang," vol. 1.
A few other usages are listed in the long article, with many citations from print.

Lighter does not give evidence for rejection of the'GP- general purpose' hypothesis.
Lighter provides no quotes earlier than 1936 (the Eugene the jeep, Segar cartoon).

The Oxford English Dictionary suggests that the 'GP- general purpose' usage derived from "Eugene the Jeep."
"Eugene the Jeep" seems to be the earliest use provable in print.

"Jeepers" and Jeepers Creepers" seem to have no connection to the usage under discussion here.