Every WWII picture and film I’ve seen shows Jeeps as open-air vehicles. But honestly, that doesn’t make a great deal of sense, given the relative ease of manufacturing a removable canvas cover, and given the fact that they had rear seats, and the rear seats of convertibles get really, really, really cold.
Incidentally, capital-J Jeep is a trademark that was granted to Willys in 1950 and which now belongs to the Chrysler Corporation (or is it still DaimlerChrysler?).
Small-j jeep is an unofficial term for any number of similar vehicles. (The Army called jeeps 1/4-ton trucks.) The ‘WWII jeep’ is generally the Willys MB or the Ford GPW (the ‘W’ signifying ‘Willys pattern’). After the war there was a similar vehicle called the CJ2, which can be identified by its larger headlamps and its tailgate. The rest was the same as the MB. Next came the actual production version (the CJ2 only being sold in small numbers) called the CJ-2A. The ‘CJ’ in CJ-2A stands for ‘Civilian Jeep’. Other civilian models were the CJ-3A (different windshield with pop-out vents in the frame instead of the earlier swing-out glass) and the CJ-5 (round fenders). Military versions were the M38 (MC) and M38A1 (MD). The CJ-3B was called the M606 and was used mostly for export. From 1959 to 1982 Ford produced a replacement for the M38-series called the M151 (series) MUTT (1/4 ton 4x4 Military Utility Tactical Truck). The M151 was replaced my the High Mobility Multipurpose Wheeled Vehicle (HMMWV, or ‘Humvee’).
For more on ‘flatfender’ Jeeps, see The CJ2A Page (which is my own creation, but has had a better message board installed since I gave it to a reader four or five years ago) and The CJ3B Page (which served as the inspiration for the former).