In terms of gaming system, GURPS would suit what you want to do better than any other system I can think of. There is a sourcebook called GURPS Autoduel that might be what you are looking for.
Otherwise adapting the Dark Sun setting and using Pathfinder rules would be pretty cool too.
A few quick points:
-Mad Max, the first film of the series takes place in the near future of 1979. Society is starting to crumble, but it largely resembles Australia of 1979.
-Mad Max 2 (The Road Warrior in the USA), and Mad Max Beyond Thunderdome, take place after the end of available oil, and a massive world war. The world as we know it has recently been destroyed, and the survivors struggle to survive among the ruins and remnants of the old world.
Mad Max Fury Road seems to take place many years after the events that largely ended civilization. It isn't clear if anyone is old enough to remember the previous society, and almost nothing remains intact that resembles the old world.
Such settings would largely resemble D20 modern, then d20 modern with severely limited tech and resources.
I would say that it would be a mistake to tie a PC to a vehicle, as vehicles come and go, and most characters in the films appear on different vehicles all the time. I really don't know what the best system for vehicle-on-vehicle RPG combat would be, but I doubt PC classes would make a good starting point. I would suggest using the monster creation rules as a starting point for making vehicles, with PC drivers "mounted" on them.
With all that said, I don't have much experience with D20 modern, vehicles, or post apocalyptic settings, so don't let me stop you if you have some good ideas. Please let me know what you come up with.
Magic breaks the setting.
Remove magic.
Thing is Mad Max is a apocalypse world that was once based on high tech. In a D+D world, there really is no high tech.
Most D+D type worlds are only low tech, so if a civilization falls, the only thing that really changes are the fall of major cities and the cessation of trade. And of course a lot of people dying. but the average lifetyle and modus operandi remain pretty much the same.
Except with the Tech Guide, there kind of is. Just use the tech guide as a source of pre-apocalypse equipment, eschew magic and casters, include only martial classes, and there you go.
For monsters you could include normal creatures that might have been altered from exposure to radiation, and after a couple of generations have become what might be found in any bestiary, though aberrations and undead would be a good fit.
I recently created a map of what I call Post Apocalyptic Motorcross Race Track. Its an area of ruined modern factory and rubblefield with broken asphalt streets, a contaminated lake, abandoned vehicles, enclosed by walls and walls of cargo boxes with a makeshift motorcross track within. I've made it available as a downloadable PDF map product on DTRPG.