The Italian Genealogical Group (IGG) of NY has compiled indexes of NYC vital records that are available on FHL microfilms. Steve Morse created a search engine for these indexes, which you can access at these links:
Births, ~1879-1909 http://stevemorse.org/vital/nybirthigg.html
Marriages, ~1864-1937 http://stevemorse.org/vital/nymarriages.html
Deaths, ~1862-1948 http://stevemorse.org/vital/nydeath.html
Although you could request a copy of the certificate from NYC Vital Records (more expensive), you can also get a copy from the LDS Family History Library:
Once you have a FHL film number and certificate number, you can either request that a copy of the film be sent from Salt Lake City to a local Family History Center (for a fee) for your perusal, or request an image of the certificate from the FHL's Photoduplication Service (free, details at https://familysearch.org/learn/wiki/en/Photoduplication_Serv...