Often, larger companies will have a specific section of their website dedicated for interns. I would recommend identifying some companies in NYC that interest you, and then seeing if they list any available internships on their website.
(There's usually an intense interview process, so I definitely recommend finding people that you know at that company and asking for a referral.)
For example, I work at Salesforce. While we're HQ'd in San Francisco, we have a New York office. Here's our webpage for college students: https://www.salesforce.com/company/careers/university-recruiting/