This proposal is a joke, IMO. Funds for IRS agents were allocated so that the wealthy, who are likely to find tax loopholes, could be audited more frequently. It's shocking to me--simply shocking-- that some politicians would like to funnel that money away from it's purpose to enact a law that has no possibility of stopping school shootings. It's pure politics.
It won't stop school shootings. It might, maybe, shorten the duration of a school shooting once it's started. Mostly, it's window-dressing.
The solution to this starts with the assault weapons ban that was allowed to expire, like Magical Felix said. If lawmakers could get that back in place, nationwide, you watch and report the numbers of mass shootings. It will decline over time. Then you can take more measures based on your success.
Limit the amount of ammo that can be purchased at any one time. Get rid of the kits that let you convert a gun to a more powerful gun. Make a longer waiting period to get guns, so that angry people have a chance to cool off. I'm a gun owner. I've always left same-day with the gun I wanted from the dealer selling it. I don't have a record but they had no way of knowing if I was mentally stable.
I like the idea of someone "vouching" for someone else when they want to buy a gun. That being said, I don't think I would ever do so for anyone. No more than I would hand them one of my guns.