I believe the SC has ruled national gun registry would be unconstitutional. I have to find the exact text.
There is federal law against it.
Firearms Owners’ Protection Act (FOPA), 1986 Under 18 U.S.C. §926, the Attorney General is authorized to prescribe the rules and regulations necessary to carry out the GCA. Section 6 of FOPA amended §926 to prohibit a registry of firearms, firearms owners, or firearms transactions. The pertinent language of §926 reads
"No such rule or regulation prescribed after the date of the enactment of the Firearms Owners' Protection Act may require that records required to be maintained under this chapter or any portion of the contents of such records, be recorded at or transferred to a facility owned, managed, or controlled by the United States or any State or any political subdivision thereof, nor that any system of registration of firearms, firearms owners, or firearms transactions or dispositions be established.
Nothing in this section expands or restricts the Secretary’s [Attorney General’s] authority to inquire into the disposition of any firearm in the course of a criminal investigation. "
Statutory Federal Gun Registry Prohibitions and ATF Record Retention Requirements
(P.L. 99-308, May 19, 1986, 100 Stat. 449, 459.)"