A couple of points.
One, I don’t think this idea will totally prevent gun crime in the US. I also don’t think it will have any effect on the spectacular mass shootings we have. While undeniably tragic, it is the day-to-day homicides which are a much bigger problem. It also wouldn’t do much about suicide, the biggest portion of gun-related deaths. That said, do you believe it would have no effect on gun-related violence? True, it would take a while before criminals’ stocks of ammunition was expended and foreign black market channels would exist, but unlawfully getting ammunition would get harder and harder. The idea that some handloader or some guy making black powder in his basement is going to supply anything other than a tiny fraction of the illicit market is laughable.
As far as being licensed to use your constitutional rights, in my state, Texas, it is required that I have a photo ID to vote. I also need a license to carry a pistol. The Second Amendment and the Heller and MacDonald decisions do not preclude registration, licensing, etc. of firearm. They say that you can’t ban guns or de facto or de jure.
As far as making this merely a state law, I would point out to you that state borders are porous. When I was a kid, I remember being stopped at the California border to be checked to see if we were carrying any fresh fruit or vegetables, and I suppose that it would be possible for a state to attempt to prevent smuggling of ammunition across it’s borders, but states like Maryland, Illinois, New York, Massachusetts and New Jersey don’t do anything like that now. I don’t know if that is for legal, political or practical reasons or some combination.