In the past I have gotten birth records from the NYC Dept. of Health for dates after 1909 if I could prove that the person was deceased and that I was next of kin. They may have changed that but here's the website.
http://www.nyc.gov/html/doh/html/vr/vr.shtmlI have found two William Grants in the Bronx, both born about 1919. Do you have any siblings' names?