Registration systems operated by the states are not governed by the FOIA.
Please note that the FOIA applies to federal agencies, and those are defined as (quoted from 5USC551)
" “agency” means each authority of the Government of the United States, whether or not it is within or subject to review by another agency, but does not include— (A) the Congress; (B) the courts of the United States; (C) the governments of the territories or possessions of the United States; (D) the government of the District of Columbia; or except as to the requirements of section 552 of this title — (E) agencies composed of representatives of the parties or of representatives of organizations of the parties to the disputes determined by them; (F) courts martial and military commissions; (G) military authority exercised in the field in time of war or in occupied territory; or (H) functions conferred by sections 1738, 1739, 1743, and 1744 of title 12; subchapter II of chapter 471 of title 49; or sections 1884, 1891–1902, and former section 1641(b)(2), of title 50, appendix;"
Please note that state government agencies fall outside of this definition.
Some states have various access systems that may work similar to the FOIA. My experience is from California where we had the "Public Records Act." In the early 80's my agency received a request from the CBS Media Organization to release records of folks who had been issued CCW licenses. We declined the request, citing a provision within the PRA that allow for the withholding of records where the privacy interest of the record subject exceeds the public interest in the record (California Government Code section 6255). CBS took the matter to court and prevailed (refer to the California Court of Appeals decision in CBS Inc v Block). As a result, our CCW permit records were provided to CBS. With case law now established, you should also plan on California CCW records being released on request.