It depends on the gig, I guess. Some jobs will require a drivers license while others won't. If a job is down to you and another guy who has a driver's license, I would expect the job to go to the other guy.
When I was 17 I worked as a runner for a concert venue. My job was to basically do all the crap work that everybody else was too important to do, and that included things like picking up dry cleaning, running to the hardware store, or picking up talent (imagine you're an internationally known musician, expecting a limo, and a dorky kid in a beat up station wagon shows up!!). Without being able to do any of that I don't think I would have gotten the job.
It doesn't hurt to apply, though, and there may be a studio position where you don't have to drive.