Of course, there are broad rights to access many government records secured under state and federal statutes, usually called Freedom of Information Acts or Public Records Acts. But those statutes tend to have statutory exemptions for private information about particular people. Whether or not the sheriff should release the records turns not on the First Amendment, but on the relevant West Virginia statute and its privacy exception.
Now, if the West Virginia statute does mandate release of such records, notwithstanding the privacy exception, one could craft a constitutional argument that the Second Amendment right to own guns includes a right to own guns free of public disclosure of such gun ownership. But I’m not sure that courts would accept this argument; see pp. 1545-49 of my “Implementing the Right to Keep and Bear Arms” article for some general thoughts related to this point (though they focus more on disclosure to the government, rather than redisclosure by the government). One could also in theory also craft a constitutional right to informational privacy argument barring such public disclosure; but that too is unclear — whether there is such a right, and what its scope might be, is unsettled.
Still, even if courts do accept such an argument, and say that the Second Amendment or the Due Process Clause right to privacy bars disclosure of gun records even when a state statute mandates its release, that wouldn’t involve the Second Amendment trumping the First Amendment. It would involve the Second Amendment (or the Due Process Clause) trumping a state statute that isn’t mandated by the First Amendment.
And beyond that, courts first have to decide whether or not the state statute even mandates the disclosure of such records, or whether the privacy exception exempts such records from disclosure. And if there is such an exemption, nothing is trumping anything else — it’s just that the state statute, by its own terms, wouldn’t be applicable to these records.