You need a driver's license to be an EMT. At most courses, you can start if you are not 18 as long as you are 18 prior to taking the certification exams. The same may also be true of your driver's license. I would call specific programs to verify this before signing up for one.
Also, just keep in mind that as companies will check your driving record before they hire you. Be careful about things like speeding tickets - they can cost you a job as an EMT even if obtained off-duty or prior to starting.