Anarchy is a political structure where there’s basically no one in charge, right? But wouldn’t that just create a power vacuum that would filled by organized crime, corporations, etc.? Then, after that power vacuum is filled, we’re right back at square one, and someone is in charge.
Are there any political theorists that have come up with a solution to this problem?


Taking the definition at its etymological root, all anarchy means is “without rule”.
In my head-canon, that doesn’t necessarily mean the lack of laws, state, institutions or governance; the implication is that there are no citizens or individuals with permanently elevated authority in the polity of government. Without rulers.
Many, of course, disagree with this mostly on the basis of practicality, but I’d like to think it’s another way to describe the concept of “No gods, no kings, no masters, no slaves.”
But then there’s the issue of who enforces the rules
Democracy is supposed to be that, but the citizenry doesn’t participate like they should so it devolves to where things are now.
Because the citizenry are disempowered, they have delegated their social obligations to institutions to handle it for them and thus hold no stake in what happens.
They view the result of politics as something that happens to them rather than something they influence, and frankly they’re right. We elect representatives who maybe hold one or two values we want, yet constantly act out of our own interests.
The only true democracy that can last is a direct democracy where everyone votes on the issues they want too.
I talk about participation in another comment, coincidentally.
The “archy” is definitely related to the idea of a top-level leadership/executive group that is set apart from the rest. Anarchy removes that executive privilege and parasitism.
For any society to function, there must be rules. Anarchy, in most forms, the community is the legislature and judiciary.
I guess you mean “without rule” as in “without people ruling” and not as “without norms”, and it is indeed correct. There is a word for “without norms”, which is anomie (at least in french).
Also, i’d argue that states and governance inherently require permanently elevated authority, but if you meant more general meaning for those, like state as organization of masses of people and governance as common decisions for those masses, then i see your point.
The “political class” of an anarchical state as I’ve described would be rotationary.
We in the United States have “Jury Duty”, where the average citizen is required by law to be selected to be part of a “jury of peers” on legal cases if the defendant exercises their right to a trial by jury.
Jurors can be struck down (relieved of their duty) for many reasons in the jury selection phase by attorneys, the judge, or submitting documentation on why they can’t perform their duty.
A corollary compulsory service or duty could be applied to the positions in the three branches of government we have in our current constitutional structure.
We would effectively shift from being a constitutional federal republic (on paper; in practice, the current form of government is a plutocracy) to a constitutional aleatory republic. We would have representative governance, but they’d be subject to review, competency approval, and votes of confidence.
One could also imagine ranked-choice voting and mandatory direct referendums regarding crucial policy decisions. Lobbyists must present their legal proposals to jurist-representatives and the general public, mitigating the efficacy of monetary influence in political speech and advertising.