I am not aware of the exact law in NY state but someone below has answered that for you. Regardless of if it is allowed or not, I want to stress that this is your face you are dealing with and you have very important blood vessels, nerves and muscles that require an in-depth knowledge of the anatomy in order to provide the safest possible treatment. Dermatologists, plastic surgeons, facial plastic surgeons and oculoplastic surgeons all get formal training in this anatomy. Many people take weekend course or are trained for a few hours by the company that sells the product. Think about it for a minute and decide who would you want treating you? Saving a few bucks on an unskilled injector can have long-term disastrous consequences.
No, estheticians and LPNs cannot inject anything in New York. Technically, an RN can inject under a physician's direct supervision but I personally do not understand why anyone would seek out a nurse over board certified plastic surgeons or dermatologists who have had extensive training in the facial anatomy. Buyer beware. I would also be skeptical about internists, OBGYNS, pediatricians, etc. who are trying to branch out into cosmetic procedures after taking a one day course, as opposed to seven years of surgical residency.