Just tell her that you don't have a car. I know people used to think that having a car was a big status symbol, but I think that is changing a lot. Speaking as a 20 something woman, if a guy was like, "Hey, I don't have a car because I use public transportation most of the time. I'd really like to take you to xyz place because it's delicious/fun/interesting/whatever, would you mind driving?" I would be totally fie with that. I would probably respect that you don't pay for a car you don't need, contribute to pollution, etc. Don't feel emasculated because you don't have a car that you don't actually need. If she is awesome, and I assume she is if you want to date her, she will not think less of you for making smart choices about your lifestyle.
No disease that can be treated by diet should be treated with any other means.