Westchester or Long Island
You can travel outside of the city and apply to the clerks office in Mamaroneck, NY. You must have an original birth certificate and a government issued photo ID. More info at Mamaroneck City Clerk
Its a 42 minute train ride from Grand Central Terminal and a short walk to the clerks office from Mamaroneck Station. The train cost is approximately $19.50-$25.50 round trip per person. You must first call them to schedule an appointment.
Babylon Long Island Town Clerk is open as well for licenses. You must have a birth certificate here as well.
Take the LIRR to Lindenhurst Train Station from Penn Station and then a Taxi to 200 E Sunrise Highway Lindenhurst, NY 11757