During the SCW both belligerent sides used everything they had at hand to transport artillery, included civilian trucks and agricultural tractors.
Before the war the military vehicle pool was of the 1920s, little and old. Hispano-Suiza, Ford and General Motors had factories in Spain and were the main suppliers of the Spanish army. Also the army had Nash-Quad, Holt Caterpillar and Pavesi tractors as artillery prime-movers and some Mercedes Benz heavy trucks and Dodge mod.1932 light trucks.
At the outbreak of the war, Stalin sent to the Republican Goverment around 10.000 vehicles included:
- GAZ AA (Ford-AA 1929 1.5 ton Soviet copy truck)
- ZIS-5 (3 ton truck)
- ZIS-6 (six-wheeled 4 ton truck)
- Komintern and Stalinetz tractors
Also arrived some french and british trucks but not in relevant quantities.
In the other side, Franco bought in the U.S.A around 12.000 trucks mainly:
- Ford V8 1936 and 1938 light trucks
- Chevrolet 1.5 ton 1937 and 1938 light trucks
- Studebaker J-15 light truck
- Caterpillar Mod. 22 tractor
The Condor Legion arrived with around 1.000 all types vehicles included around 300 trucks and 170 tractors as:
- Krupp-Protze Kfz-69 (six-wheeled 1 ton light truck used to tow Pak 36 3.7 cm AT gun)
- Krupp-Protze Kfz-70 (six-wheeled 1 ton light truck used to tow Flak 30 2.0 cm AA gun)
- Mercedes Benz G3 (six-wheeled 1.5 ton light truck)
- Opel Blitz (1 ton, 2,5 ton and 3 ton trucks)
- Henschel 33 G1 (six-wheeled truck used to tow Flak 18 8,8 cm AA gun)
- Mercedes-Benz LG63 (six-wheeled 2.8 ton truck)
- MAN E-3000 (3 ton truck)
- VOMAG type DL 48 (8 ton heavy truck)
- Büssing-NAG Type 80 (11 ton heavy truck)
- Sd.Kfz. 7 KM7 and KM8 (8 ton halftrack used to tow Flak 18 8,8 cm AA gun and leFH18 10.5 cm howitzer)
The Italian army sent around 6.700 vehicles included:
- Fiat 618 (1.2 ton light truck)
- Fiat SPA 38 R (2.5 ton truck)
- Fiat Dovunque 33 (six-wheeled 2.5 ton truck)
- Fiat 634 (7 ton heavy truck)
- Issota Fraschini D80 (5 ton heavy truck)
- Ceirano 50 CM (5 ton heavy truck)
- OM-32 Autocarretta
As the spanish road network was very bad and the vast majority of the field guns used were horse artillery in origin with wooden wheels, the guns usually travelled on the flat beds of the trucks when these were available.
Here you can find more info in spanish about this topic:
link 
paints up some WWII Soviet aircraft.
