New York does not have a court procedure for filing for emancipation. Some states do, but New York does not. Instead:
the child may also self-emancipate prior to age twenty-one, thereby terminating her right to support. The self-emancipation doctrine has its origins in the fact that children achieve majority at age eighteen, thereby terminating parental custody and supervision, but are ordinarily entitled to child support until age twenty-one. The courts have reacted to that dichotomy by developing the self-emancipation rule. The “adult” post-eighteen child is perfectly free to emancipate; the price is a waiver of child support.
Merril Sobie, Practice Commentaries, McKinney's Cons Laws of NY, Family Court Act § 413