NYS laws regarding private gun sales.
State law requires people buying or otherwise obtaining ownership of a handgun at a gun store or through a private sale to possess a license, which requires applicants to go through a criminal background check. In addition, a background check is required on private sales of all firearms at gun shows. Private sales of rifles and shotguns at places other than gun shows, such as through newspaper advertisements, or the internet are not subject to a background check requirement.
You’d have a tough time suing the seller and pretty much an impossible time suing the site considering NYS law doesn’t require any kind of background check on a private gun sale for rifles and shotguns.