I would say yes with stipulations, as I can understand that this can be a way to funnel firearms into the hands of law abiding citizens and away from criminals. Off the top of my head I would worry that this could be turned into a "no fly list" for potential gun owners. What would be the information used in the background check and who would be in control of that information? Also, I'd like to see a built in notification system for anyone who is disqualified.
Oh, thou clear spirit, of thy fire thou madest me, and like a true child of fire, I breathe it back to thee.