No. Read Machinery of Freedom by Friedman for a primer on why this is not necessary.
It requires a utopia to work as described. Libertarians love it.
I'll... leave judging the soundness of that plan as an exercise for the reader.
[edit, also:]
> The plausible end result of violating the law is either that you have to accept the result of arbitration or you end up getting shot
You would contract with a particular for-profit legislative system the same way you contract with an insurance company and a police company and a war-defense company. The insurance company would have contracts with investigative companies and retributive force companies.
When you get into a dispute with your neighbor over whether they are allowed to build a wall that might cross on to your property, you contact your court company and your insurance company. The court company sends someone to talk to your neighbor and get details of their jurisprudence subscription. If they subscribe to the same courts, excellent! They have jurisdiction and can decide whether this is a minor violation of the contract (see rules on respecting property lines) or a major violation. Violations can be punished by corrective action or fines equivalent to the costs of such.
If the neighbor uses a different court system, perhaps they have a treaty-contract with your court system. In that case the treaty-contract will supply the rules.
Oh, but maybe there isn't a treaty-contract. The court companies might take this opportunity to negotiate one. Or, they might notify their insurance companies that they have a conflict. Maybe their insurance companies have a mutual non-aggression contract, in which case they might force the court companies to negotiate a treaty-contract or lose insurance coverage.
See? Utopia!
There's a long quote down a ways (starting: "How, without government, could we settle the disputes that are now settled in courts of law?") that does a pretty good job of giving you the gist. The very TL;DR, and I don't think an unfair one (read Friedman's own, quoted story of how his system might work, if you don't believe me) for anyone who's familiar with cyberpunk fiction is that it's basically just unironically advocating the corporate-security-is-the-only-security future that those settings often feature. Given the timing, it seems probable to me that Friedman and others' writings on this topic inspired those elements of that genre, in fact.
In Friedman's version of the story, this looks like a lot more freedom than we have now[0]. In the cyberpunk version, it looks a lot like feudalism.
[0] I don't really see that, personally, in his and similar writing, which looks like it replaces de jure restrictions with a pile of stress, anxiety, and risk, which don't really feel like more freedom in any practical sense, to me, but it's supposed to be there, so let's just grant for the sake of argument that he's successfully painting a picture of a more-free system, as that's what he intends.
These authors comment on the idea that war being expensive would be a barrier, and surely it would be some kind of barrier. On the other hand, during the late medieval and early modern eras there were numerous wars that easily cost the entire revenue of the "state" for the year or sometimes multiple years. These wars were privately financed and involved the use of mercenaries. To me, this doesn't sound so different from the world proposed by libertarians like this, and I much prefer our world.
"We hold that rights to property, including real property, are individual rights and, as such, are entitled to the same protections as all other individual rights. The owners of property have the full right to acquire, trade, control, use, dispose of, rent, collateralize, or enjoy their property in any manner, without interference, unless the exercise of their control infringes upon the valid rights of others."
Does the Libertarian Party of Colorado supply some sort of hierarchy to these platform elements since some of them conflict.