Not federally. It's up to the states. Many require a handgun registration. I don't think any require shotguns or standard long barrel rifles to be.

It's a gay bill that is dead in the water.

“Permit to purchase”: Ten states have enacted permit to purchase licensing schemes that require prospective purchasers obtain a permit or license before buying at least some firearms.12
“License to own”: Three states—Illinois, Massachusetts, and New York—require a license to own firearms (New York’s law applies only to handguns). Unlike a permit to purchase, a license to own a firearm must remain valid for as long as the person owns the firearm.
Firearm Safety Certificate: Two states—California and Washington—require prospective firearm purchasers to first obtain a certificate showing that they have completed required firearm safety training. California requires this for all firearms, Washington requires it only for semiautomatic rifles.
Registration: One jurisdiction—the District of Columbia—has a registration law that also functions as a license requirement.