As far as I know, there aren't restrictions on ammo purchases. I've never once been asked to show any ID, or have a background check on ammo purchases.

As a side note, there are also no restrictions on reloading supplies or machines. Literally anyone can purchase the materials and presses to make their own ammo.

I bought a press for my Dad for his birthday when I was 16. Walked in, paid, walked out. No big deal.