Depending on the US state you live in, an online ministerial license may not be enough to allow you to legally marry someone, as the state may impose additional requirements - ie, must be a state-certified/elected Justice of the Peace, etc.

This is because marriage licenses are a governmental/legal document with specific legal meaning (in particular when it comes to taxes, medical/hospital visitation rights, etc), so it's not exactly "meaningless".

Because of the legal ramifications of the marriage license, the state has to recognize the agent certifying said license, otherwise it's not legally binding.

I believe that you will find that most laws reference an "officiant," not necessarily a minister.

This is probably more theory than you want, but there's an idea that marriage is the sort of thing that lies at a crossroads between personal and societal needs. Anybody can shack up, but marriage provides a sort of stabilization factor (and historically this is where you see appeals to the "production of legitimate children") that the community at large has an interest in promoting, and does in part by attaching a whole package of rights (inheritance, divorce, et cetera) to it.

These are powerful rights, and potentially subject to abuse, and ones that the state is going to be asked to enforce, so the state, in theory, wants to know that there's someone it can trust to oversee it, with the additional idea that it's supposed to impart the solemnity and importance of the act. There's someone else there, and probably witnesses too, to make sure that the parties understand that this is a Big Deal, and there's got to be at least the nod to ceremony.

And, of course, historically (and still in some jurisdictions) you don't need an officiant or a solemnization, because common law marriage still exists, where you were married if lived together long enough or held yourselves out as married. But that was treated as something of a patch, in the U.S. at least, for when the population was more sparse and travel was harder (or so the story goes. It's just as likely the difference between small clusters of tightly knit societies and larger ones) and you can see how it might cause trouble - any time surviving spouse and family don't see eye to eye (and they typically don't), there's going to be a court fight.

I don't know the history of the 'minister licence industry,' but it does predate the internet.

Sorry, I'm removing this as it is too current-events. If you're interested in when marriages became under state control please feel free to reword your question to make it more explicitly historical and resubmit. If you're curious about state-recognized marriages today, try /r/AskSocialScience where the sociologists hang out.