I expect the test is for lead, which I believe is required for NYC kids in day care or other care or school settings. We had to do it when our twin daughters were 11 months old, before they first started to go to daycare. I don't think it's possible to get enough blood to do the test properly with just a finger stick.
[UPDATE ADDED LATER: Since I see several responses from people balking at the test being mandatory, I thought I would check it. Here is a document from the NYC Department of Health that says it is New York State law to test all kids for lead at ages 1 and 2: http://www.nyc.gov/html/doh/downloads/pdf/chi/chi26-3.pdf
It also says only about 70% of NYC kids are actually tested, which may explain why some here haven't heard of it or are surprised. It also says, "Venous specimens are more accurate than fingerstick specimens. Environmental contamination of fingerstick specimens can result in false positives, and finger squeezing can dilute blood and result in false
negatives."]
We took our girls to a Quest and asked for someone experienced with little ones. Still, it was traumatic. In our case, there was a lot of screaming and several attemps to get the vein. BUT they were perfectly fine soon afterwards. It's just a trial you have to get through. The Quest people let us use a private room right after so I could nurse my daughters and help calm them down. If you're not nursing, I'd recommend bringing a bottle or something else soothing for after the draw.
Fwiw, the second-year blood draw was much easier.
Good luck,
A.