Actually, in the case of school shootings, most of the criminals get their guns from either relatives (who get them through legal sources) or through legal means themselves.
I'd be more concerned that this is a case of closing the stable door long after the horses have bolted. Automatic weapons are already out there, and people will not want to give them up, even if they cannot justify owning them in the first place.
It really does need to be borne in mind that, when the Second Amendment was created, gun firing rates were at a level where mowing down a crowd of people was not even conceivable, I'm sure even the most Freedom loving of the Founding Fathers would have blanched at the concept of just anyone running round with weapons with that kind of power.