Now that is an interesting prospect that might work and right, theoretically not against the constitution. NRA's lobby is sadly too strong for it to come off the ground though.
Here's a better solution. Seeing as 90% of gun crimes are committed with stolen weapons, how about imposing life sentences on possession of a stolen firearm (plus a requirement to register stolen weapons with the police)? This wouldn't infringe on private sales as anyone can make a call to the police station and perform pre-buy due diligence. The NRA supports this position.
That would make selling or receiving a stolen weapon much more risky, and ultimately drive up the price. The laws on the books right now for possession of a stolen firearm are more or less a slap on the wrist.
How about life sentences anytime a gun or other deadly weapon is used in a crime? The NRA also supports strong sentences for people that break the law.
Enforcing and strengthening exisiting laws seems a much more logical approach than putting new ones on the books.