No rule of law means the people become vigilantes as their government fails them.
The reason vigilantism isn’t likely is pretty mundane. It requires skills the average guy doesn’t have, as well as the ability to track and find someone which the average person also does not have. If you look at the vigilante movies, “Taxi Driver,” “Vendetta,” “John Wick,” they all have means and opportunity. The reason average people don’t run around solving crimes, aside from getting arrested for interfering with the police, (Or, whatever they want to charge you with to make you stop) is because we don’t have means, opportunity or motive.
Trust me, the police would never stand for someone like Sherlock Holms or Batman or even an average Joe solving or preventing crimes because they don’t like the bad comparisons that would result. If anyone did those things you can bet the government would be much more interested in preventing them from being successful than it is in solving or prosecuting the crimes the “hero” is resolving.
Every time Commissioner Gordon turned on the Batlight it was an admission that he’d failed in his job and his whole department needed to be replaced by a guy in a cape. A real commissioner would rather the entire city burn to the ground rather than make that admission.