A number of indexes are available to locate these records:
Reclaim the Records has indexes available through 2017. However, records less than 50-years-old are not public records, and cannot be ordered. For records from 1938 to 1969, you can order the records from the New York City Clerk's office: https://www.cityclerk.nyc.gov/assets/cityclerk/downloads/pdf/mail_in_request_form%20final.pdf
I've worked with these records on numerous occasions.
Best of luck in your search,