Can you start a business in New York without a green card?

No, you do not need a green card to start a business in New York All you need is proof of legal presence, and that is your social security number (and in some counties, your work visa / authorization) ... The key is to pay your taxes any income you receive from your business

Go to the website of the City of New York, the local government about starting a business, including tips and tricks http://www.nyc.gov/html/sbs/nycbiz/html/