Which article of the US Constitution allows state laws to prohibit the implementation of gun laws

The US constitution in general doesn’t list things that states are allowed to do, it sets up the Federal government and puts some restrictions on states (like not allowing them to conduct foreign policy). States are considered sovereign and can pass any laws that aren’t forbidden by the constitution. Since the constitution doesn’t restrict how states manage their subdivisions other than requiring a republican form of government and forbidding some types of discrimination, it allows them do do pretty much whatever they want with them. States decide whether counties, cities, towns, or whatever exist, and what powers they have. In some states, especially those like NY and IL where there is a single really large city, there’s a lot of local governing power. In other states, local governments can’t pass any criminal laws.

I’m not really sure what legal theory you have that would prohibit states from passing preemption laws, but they’re something that’s been around pretty much forever and the concept is completely uncontroversial.

Bullfrong County, Nevada was an interesting take on the question. The US government was going to put a nuclear waste storage facility in Nevada, and the legislation to do so gave money directly to the county, not the state. Nevada responded by creating a new county that had no residents, covered the area where the facility was to be built, and set property tax rates to the highest allowed under state law. This would have redirected all of the federal money that was earmarked for the county to the state. It also created some problems, like the fact that the county didn’t have any residents so couldn’t elect any officials (they were to be appointed by the governor) and could not create a jury to try any crime committed in the county (NV law requires a jury from the county where the crime happened). It never got to the federal level because the state courts ruled that the lack of republican governance was a violation of Nevada’s constitution, but I suspect that if the state allowed the county to exist then the Feds would have taken it to a Federal court and it would have been ruled invalid for the same reason.

Another case where the “republican form of government” clause has been relevant is in the setup of state legislatures. At one point, some states had a bicameral legislature modeled after the federal one: A state house of representatives where each district had the same population, and a state senate where each county had the same number of senators, regardless of population. But the courts have ruled that this isn’t republican, and that each legislator in each house must represent the same number of people. Most states still have a bicameral legislature, but with less difference between the two houses (usually, there are still fewer senators, and they still serve longer terms).

The problem wasn’t that it wasn’t republican, but that it violated the equal protection clause of the 14th amendment. That was the basis of Reynolds v. Sims.

The practice is typically called preemption. The preemption law in FL is as follows:

My understanding is that the vast majority (>40) states have some form of preemption with varying degrees of breadth. Here is a listing by state that summarizes, with relevant statutes linked.

It’s called “local preemption.”

IANA Constitutional Scholar, but Google tells me it’s the Supremacy Clause in Article VI

It reads, uhhhm:
“This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.”

Do federal anti-corruption laws operate under the principle of guaranteeing a republican form of government or some other authority? Has anyone ever petitioned the federal government to intervene when state election fraud was so rampant that it could be claimed that a republican form of government was no longer in operation?

Your response is completely unresponsive to the question. The Supremacy clause has nothing to do with what we are talking about. It relates to issues where state laws conflict with federal laws, or the federal constitution.

My Constitutional Law professor, the estimable Charles Alan Wright, merely told us that the “republican form of government” clause has never been given any meaning under any Supreme Court decision, and thus is essentially a meaningless flourish at this time.

I seem to recall that there are some nineteenth-century cases (revolving around the post-Civil War administration of the defeated Confederate states?) in which the Supreme Court held that the guarantee of a republican form of government was a political question, and therefore one on which the Court would defer to Congress/the President.

A little-remembered case from a little-remembered pre-Civil War uprising: Luther v. Borden - Wikipedia

In a democracy, what is not prohibited is permitted.
In a dictatorship, what is not prohibited is compulsory.