Here are a few relevant links:
A 2005 article about stunt drivers.
An apparently defunct stunt driving school (not updated since 2006).
Another stunt driving school. Check out their video here.
So how to go about starting a career? I think a good, and not too expensive, place to start would be amateur motorsports, like autocross, high performance driver’s ed, or amateur rallying or racing. I started a thread about HPDE here. Spend a year or two driving on track in your spare time, developing basic car handling skills. You don’t need a very fancy car, and you can take it at your own pace, and at a level of expenditure that you can manage while holding onto your day job. And you’ll have a lot of fun doing it, even if you never become a pro driver.
If you’re considering a stunt school like in the links above, you’ll get a lot more out of it if you come to it with some track experience. An HPDE school may cost you as little as $200 a day, compared to over $850/day for the stunt school.
When you have some skills, start looking for a way into the film industry. This is much trickier, and there’s no simple or guaranteed way in. It’s a matter of luck and who you know. The people you meet at the track might be able to help you network. Try to find local filmmakers or ad agencies who might need an on-camera driver, so that you can start building a reel. You might even just get a buddy to film you doing some (legal) stunts (in a safe location) to have a video that shows what you’re capable of. Once you get a foot in the door with a real job, things may get easier.
But as far as actually making a living at it, that’s going to be a hard road. Automobile advertising is one one of the most high-budget, high-pressure industries there is, and the competition is fierce. A film shoot can cost hundreds of thousands of dollars a day, so the producers make sure that everyone on the crew is the best they can possibly be, so as not to waste precious time. I’d wager there’s a relatively small group (a few dozen, maybe) of top pros who get the lion’s share of the major auto makers’ work.
Of course, that’s the top tier, and there’s lots of other work to be had, but widely spread out geographically and chronologically. Is there enough to build a full-time career? That depends on how much you need to live on, but I doubt there are a whole lot of people making a full time living at just driving for movies/TV/commercials. IOW, you may have to do other things in addition to the driving, just like most actors wait tables.