FMLA is a federal law that requires employers (with more than 50 employees, I believe) to offer unpaid leave for family or personal medical issues WITH job security. So, if your new company is covered under the law, they have to offer it.
http://www.dol.gov/whd/fmla/employeeguide.htm
I would not ask the new job about it as they may withdraw the offer.
I just checked the guidelines and you have to work for 12 months to be eligible. What I would do is say to the new job, that you have an autistic son, and that once a month or so, you need to take him for an appointment, but that you will make up the time in other ways.
This is really not covered under FMLA as it is more for long term leaves. If you made a good enough impression during the interview process, it would seem to be a reasonable request.