Anarchism is not a form of capitalism; anarchism is a form of collectivism, where individual rights are subject to the rule of competing gangs.
Whatever you are proposing it is not capitalism. The proper name for what you purpose is not ‘corporations’, but street gangs, who fund their business through extortion of their victims (which you would call ‘customers’).
Government is an agency with a monopoly on the power to legally use force in a specific geographic area. What you purpose are multiple agencies (what you incorrectly accuse to be private corporations) in the same geographic area, that have the power to use force subject to no rule of law: anarchism.
Under capitalism, corporations are the result of a specific contractual legal framework (provided by government), based on the principle of individual rights. Without government, the distinction between public (state owned) and private no longer exists. Corporations cannot exist without individual rights, and governments to protect those individual rights.
[However, keep in mind that corporations are not creatures of the state, no more then individuals are.]
Those who attempt to combine anarchism with capitalism, make the error of confusing the peaceful form of competition of capitalism — trade, ideas, and dollars — with the brutal “jungle” form of competition of anarchism — brutality, whims, and bombs.
Have you ever thought what happens when one ‘corporate protection agency’ disagrees with another? By what method do they solve their dispute? They do it by competition not with dollars, but with guns. They seek to solve their dispute by resorting to force against each other, i.e., a perpetual state of civil war. Under such a system, which gang wins? The one that is the most brutal.
Anarchism is not a form of capitalism; anarchism is a form of collectivism, where individual rights are subject to the rule of competing gangs. Under such a system, any individual would beg to be placed in the relative safety of a dictatorship.
The only peaceful solution to such disputes is to have one agency with the power to settle those disagreements, according to one set of objectively defined laws — a government. This is what corporations do under capitalism, when they have a dispute with each other — they go to court (government).
Those who advocate anarchism seek to replace a rule of law, with a rule of the jungle. The kind of ‘corporations’ they envision are not corporations like those under capitalism (which have no power to resort to force), but outlaw ‘competing governments’, i.e., gangs.
For those who want an illustration of what happens when two ‘competing-governments’ are arguing with each other in the same geographical area, see: Bosnia. This is the result of the anarcho-capitalist’s ill-thought out nightmare: a species of collectivism, where one is subject to the whims of the tribe or gang in power.
On a micro-level one can observe anarchism in black markets, where drug dealers compete with each other on the same “turf” to “protect” their interests. It is to subject “might” to “right”, that one requires rights, and that one requires a government to protect those rights.