Sort of. There is a registry, inspections, and a pretty involved process to transfer ownership.
Plus a gun on its own it nearly useless, yet bullets on their own are almost lethal. We should be freaking out less about guns and spending some time coming up with a thoughtful and practicable way to track ammunition purchases and sales more closely.
Like did you know that I, with my dinky 100 year old target rifle in 22LR can walk into an ammo shop and buy 1000 .308 human killing rounds, then another and do it again. There's no requirement I own a gun that can shoot them, no limit on how much I can buy, and sales tracking isn't shared between ammo shops.
By the time anyone has figured out what's going on I can have gone and done something extremely anti social.