All the longer trails in Harriman have shelters (Long Path), Ramapo-Dunderberg, Suffern-Bear Mountain, however, it is technically illegal to camp in Harriman, it's only legal to stay in the leantos. I'm not sure whether they even will allow tenting around leantos... Just letting you know.
Suffern-Bear Mtn has some really nice stone leantos on it, Big Hill and Stone leanto come to mind. There's a metal leanto on the LT that I can't imagine being to nice in winter or if it's raining..pitter patter... pitter patter!
If You want to just practice winter camping, heck, you can do so in your backyard!!!! Assuming you have one and there is enough snow in Long Island...
Jay