My NJ license number is not working for the New York refund. What should i do?

I would think NY is looking for a New York drivers license, so this doesn't surprise me.
you should see an option, "don't supply", or "don't have a license"

Make sure you are just entering letters and numbers, do not use any special characters. Your driver's license number is not  required to file your taxes. However, in some states, omitting this info may delay the processing of your return.