I don't drive, but I do have friends that would help me out in an emergency. They are all about taking care of kids and animals, as am I.
Not driving will prevent you from being a traveling sitter, but it should not affect your ability to be an in-home dog host and board dogs in your home. I would make very sure that you have transportation available, however, in case of an emergency if you have to take a dog to a vet.
Depends on whether or not you will need to drive as part of the stay. When I lived downtown here I didn't have a car, and I was able to board. I did have a Car2Go membership which I could use in an emergency (and there are some dog-friendly taxis around here), but I was also within a few blocks of multiple vet offices. It may be harder to be a travel sitter depending on the public transportation options near you, but it's still possible.