NYS state absolutely does bar automatic weapons (except with special licenses - and they're nearly impossible to get). Pistols are darned hard to get, too.
I didn't check my BATFE 'big book' on NY'S weapons law, but I note you say "Special License" required. Where they are legal 'the special license' is pretty much standard. And yes a lot of paperwork (red tape) is involved and money for the transfer tax(s). But for people who have the time, money and WANT that full auto, they'll do it.
The thing is I just don't like the tone of this article. If it was a bunch of white guys, they'd likely be called patriots. This thing is just bigotry IMO.