I'll ditto Savage Worlds SWADE - I've run many a car chase and a battle using those rules, with very good results. Vehicle battles using miniatures on a tabletop with SW are also doable and I've done a fair amount of that with tanks and AFV in Weird Wars campaigns. They're a bit more doable using the vehicle ACCeleration stat in the Deluxe edition of the rules, which was removed for SWADE. There's a companion titled Road Rage, Life in the Fast Lane for the Interface Zero 2.0 (Cyberpunk SWD setting), which provides details for running vehicle races with a small section on combat. Most if not all of that content can be used with SWADE.
The Cepheus Engine could also be a good set of rules for running car battles, with the 1 caveat that careers would likely need to be added to or edited to better support PCs with good driving skills. That could also be achieved to a certain extent by modding backgrounds. There's a number of different flavors of those rules and my own personal preference for vehicle-focused adventures is Moon Toad Publishing's edition. Mostly because they also publish an excellent Vehicle Design Guide companion to compliment the core. There's also a freebie Vehicle Design Guide from Samardan Press, but IMO MTP's guide is superior.
