NYC nanny here!
I think it's pretty easy to find a nanny job in NYC that doesn't require driving.
I used Craigslist and created an extensive ad with lots of info and pics to attract prospective employers. I took advantage of HTML to make the ad aesthetically pleasing, and included a great deal of information that allowed readers to generate a fairly decent picture of who I was, what I was capable of, what experience I had, etc. My ad mentioned that I didn't have a license as well, but that I knew the city well and could get around easily via MTA.
My ad certainly did stand out from the shorter nanny ads found on CL because it was very long and detailed. But the various colors, fonts, and pics (of the children I use to watch- with the parent's permission of course, and other bright, kid friendly pics from Google) made it bearable to read, lol.
I got plenty of responses, and every one who contacted me appreciated my detailed, informative ad. My current employers actually found me through said ad.
I also tried care.com and sittercity.com, but I had very little success with these sites.
So, my advice to you would be to use CL. Aim to create a thorough, detailed, informative ad that includes things like why you enjoy being a nanny, your experience, education, childcare philosophy, what you're like as a person, what duties you perform (including non-childcare tasks if you're into that), what skills you have, if your looking for a FT, PT, Long Term, Short Term, or Live-in position, etc. Utilize HTML to make it aesthetically pleasing (CL has a help section on this), and include pics of yourself and, if granted permission, some of your ex-charges.
At the end, say you'll provide references and your resume upon request.
Do what you can to make your ad stand out from "all the others" (glance at a few of the other nanny ads on CL to see how they look). Give your ad an attractive title, and then hope for the best.
Good luck!