List of vehicle building games
Previous efforts to list many vehicle building games:
a
Using slime blocks, blocks of redstone and sticky pistons, one can assemble a contraption on Minecraft that can move on its own. A Google query for YouTube videos can bring many interesting insights. Exploiting duplication glitches, these machines are adapted to bomb large parcels of terrain too: see this YouTube query for examples. Furthermore, command blocks may be used to spawn flying or walking things. Regarding mods, there are many: check Zeppelin, Archimedes' Ships (archive), Davinci's Vessels (aka Archimedes Ships Plus) and Valkyrien Warfare (aka Valkyrien Skies).
b
With the aid of mods, it is possible to build vehicles and fly them on Minetest, and it seems that's actually a core mechanic on the Pandorabox server. To exemplify, see the mods Airship, Meshnode, Flyingships and Jumpdrive.
c
As BeamNG.drive requires third-party programs to create new cars and other vehicles, I'm stretching the meaning of vehicle building games to include potentially any game with modding capabilities, since the creation of vehicle assets naturally occurs in the process. I'll leave it here for now, since the game and its community revolves around creating new vehicles.