Any "profile" they create from social media posts and Internet search histories will be unconstitutional.
Unless that "evidence" includes threats of violence, inciting violence, or contradicts their responses to the 4473 questionnaire (like pot use), posts are covered under the First Amendment.
If the gov't denies a gun permit because of your posts with no specific, articulable suspicion you have committed a crime or are planning to, they are denying a Constitutional right without due process.
It's also an invasion of privacy. Imagine a gay gun buyer who hasn't yet come out with his sexual orientation, or worse -- is in the military and is gay. Allowing access to his private posts, searches, messages, etc. could have much deeper consequences than suffering another federal gov't intrusion into one's background.
If this passes, I predict tons of people in NY will be availing themselves of anonymizes to hide their online activities.
Check out Iron Key if interested. Unless you store your passcode on their server, there's no way for law enforcement to get to your Internet history even with a warrant, assuming you always carry and use the browser installed on the device. All temporary files, cookies, browser history, etc. are kept contained on the USB drive that's protected with hardware encryption. The gov't can't even crack the thing open to get at the circuits without destroying the thing, because it's filled with epoxy.
https://www.kingston.com/us/usb/encrypted_security/IKS1000