Yes. Private DNA testing is generally available and lawful throughout the United States.
New York has unique laws surrounding the conditions under which testing may be performed, though. There are several hoops to jump through if you do this in New York -- using a licensed testing agency, obtaining approval from an agent, etc.
Getting an order from a court is one way to satisfy the agent-authorization requirement, but you can also get an authorization from your lawyer or doctor. What the point of this is, I don't know, but it's the law.
If you want to avail yourself of this option, just call a nearby testing facility and they'll walk you through the process.
If you don't, just drive to another state and pick up a test from a drug store.