I am not sure what you need or where you get it. Next time I might start at the NY Police headquarters. I believe you do need a permit because I was stopped from taking a photograph near Times Square by a policeman. He said I could take pictures, but not use a tripod. And I certainly was not in anyones way that night. I was right beside a building - out of the flow of people walking by. But, it made no difference.
You can easily get a permit for using one in the Art Museum up by Central Park. There is a desk near the coat check on the lower level that you need to stop at. Only took five minutes.
Good luck.
Let me know what you find out because I will certainly be going back.
Another time I was using a tripod outside a camera store doing a demo in the city and no one said anything to me about it. So it might depend on where you are downtown. I have to guess that many places off the beaten track you might get away with it. My guess is taking pictures of some buildings is more sensitive then other ones.