Where do you live? I mean, do you live with parents and never have the place to yourself? Usually I just invite girls to come chill at my place, and if they are interested in me (as you say she is) they usually show up. Again, I live downtown in the city (and have a car) and I have what I think is a pretty sweet place
but if you want to spend time with her that's easy and doesn't require you to drive.
Also, you can make it casual to minimize the forward-ness of it such as: "some of my friends are coming over to [activity], and I feel like this would be a good chance to hang out. I would like it if you came". That way it's not "hey, come over, it's just me here
".
No doubt about it, girls like it for guys to have a car, but don't feel as though you
have to have a car. If you can't pick her up tell her she needs to pick you up if she wants to hang.
If she wants to hang.
I'm going to take on good faith that she is interested, and if that's true,
THAT is your angle. It shouldn't be on you to get a car because of her, especially if she wants to hang out with you. When you meet a girl always make the case that you're cool with where you're at even though she may not be. In effect you want her to meet you halfway, or at least find out what her dealbreakers are. Some people, as SpiderVille points out, are materialistic, and not having a car may be something she just won't put up with.
The other side of this coin though is that girls traditionally, in my experience, erect roadblocks. From what you've described it sounds to me like she's teasing you a bit about you being automotive-ly challenged. Don't take this seriously because it is not a reasonable excuse not to date you. Girls will instinctively try to walk all over guys they like. So if she gives you a hassel about a car again just be like "well, why don't you drive me then?" (playfully). This will put her on the spot a bit, and you'll know from her response if she's worth pursuing.