I can't directly attest to that (never bought an illegal gun) but from my understanding, yes, people have no challenge obtaining illegal guns.

However, you really don't even need to do that. You could just drive across the NY border to a state with looser gun laws, buy one there, shave off the serial number, and bring it back to NY. You could also just steal a gun from one of the many Americans who already own one.

You can also legally buy an unfinished lower receiver in many states (the part of a gun that is typically serialized). Since it's technically unfinished, it doesn't require a serial number. Then you drill a few holes into it and assemble it with off the shelf, also un-serialized gun parts.