There's a hitch to that. It is voluntary: if you voluntarily apply for a permit, you voluntarily give over your SSN. Nobody's forcing you to get a permit, so nobody's forcing you to give over your SSN.
Not my view, that's just how the government thinks.
They must tell you it is voluntary. This was just decided in United States District Court for Eastern District of Pennsylvania Stollenwerk vs. Jeffrey Miller, et. al. and the Pennsylvania State Police can no longer require a person give their SSN. Also, local sheriffs must issue and new permit that does not include the SSN if requested.
It all began when Mike Stollenwerk purchased a gun in PA and refused to give his SSN. He was denied and the some law enforcement people showed up at his home to check on him. He filed suit, lost in state court, but won the appeal in federal court. Contact me privately and I will email you the court documents.