In New York,
if your boat has any motor at all, even if it's not used for primary propulsion, it must be registered. You have to go to DMV and fill out a bunch of forms including one that shows proof of ownership. They do not mention what substantiates proof if you've built the boat yourself.
Whether it's a rowable dinghy or a square-rigged tall ship, it's the motor that matters. No motor, no worries. Just do be exceptionally careful if you even think of sailing in the East River. Do not attempt it without knowing exactly what you're doing and your cardboard boat is truly sea worthy. It looks far easier than it is. There are some vicious currents and it takes a skilled sailor to know how to navigate it safely. I don't know your experience level with sailing so please don't think I'm being patronizing. Sailing around a secluded bay should be fine, once you're in a major waterway, you must learn the rules of the road and how to sail under any condition you might encounter. Take lessons if you can. There are good sailing schools throughout the city and, better still, they know NYC waters inside and out. There really are some dangerous areas, particularly around the aptly-named Spuyten Duyvil and Hell Gate.
Talk to wowniceone about what he recommends. The man knows sailing far better than I do.