Here in California, the Liberal law makers have already passed a bill for Ammo background checks, will start the first of next year,. Oh and a $50,oo fee!
That part of the law (background check) takes effect July 1st, 2019.
What hasn't been determined, yet (last I looked), was how this is going to be implemented.
I read somewhere that instead of a $50 fee for a background check and 5-year ID card, it was being considered to create a $1 fee charged with each purchase of ammunition at the point of sale (unless the ammo was being purchased with a firearm, in which case the background check for the firearm purchase would suffice). That makes more sense, when you think about it, because someone's status could change with any 1 (or 5) year period, due to various incidents that might occur (being served with a DV/TRO, GVRO, an involuntary commitment for mental health problems, adjudicated by a court to be mentally incompetent, etc).
A $1 fee would generally be less egregious, too, unless someone normally buys ammunition more than 50 times a year (meaning more than once a week throughout the whole year).
In answer to the OP's question? Who knows?
Now that CA has crafted such a law, though, and got it passed as an initiative (even in a state with 8 million known, lawful gun owners), it wouldn't surprise me if other states tried to follow suit. If that happens, then it might pick up some momentum at the federal legislative level, and might pick up enough votes to make it out of committee, or even get to the president's desk. Dunno. Probably not right now, but who knows how public sentiment will evolve?
I remember when I was a youngster and my dad could order old military surplus firearms out of a magazine ad, and have them delivered to his house through the mail, without any registration or anything else required other than that his check or Money Order cleared. Look how much the laws have changed since then.
Don't mistake my comments to mean that I voted for the initiative here in CA, because I sure as hell didn't.