The JSTOR site says "Libraries and organizations license full-text content for their patrons, and if you are affiliated with one of these institutions, you can access those complete articles."

Their site seems to say anyone with a uni affiliation.

The public already paid 99% of the money for public research, yet only 1% of them have access to the fruits because those who paid 1% (for maintaining an archive) want it this way.

Information shares all of the downsides of the public commons: because it is freely available, no one wants to pay for the maintenance costs. JSTOR charges fees for access to cover the ongoing maintenance costs associated with storing access to thousands of journals and millions of articles published over several centuries. Note also that JSTOR provides assistance with locating articles relevant to the user's needs (i.e., library functions), and such services are frequently more valuable than its archival functions.

A wiki will not suffice to maintain access to this information; Wikipedia and Wikileaks have shown that. You would end up with all of the articles but no practical way to find the particular article or articles you are looking for.

Do you know of anyone working through the legal system to effect such change?

That doesn't follow.

https://en.wikipedia.org/wiki/JSTOR

We are not talking about "authors" in the sense of some guy living in a townhouse, smoking a pipe and drinking scotch while writing a novel. We're talking about people who are paid to do scientific research, usually by the government, and who must publish their work in a journal or conference to continue continue or advance their careers.

Really, JSTOR is not even that bad; they are a search service. Look at the journals themselves if you want to talk about a scam: the authors are not paid by the journal publishers, the journal publishers do not fund grants any more than any other tax payer does, and it is often the case that the reviewers of the articles (peer review, the foundation of scientific publishing) are unpaid, and in some cases even the editors of the journal are volunteers. What do the journal publishers do? They print and bind journals, sometimes, or else they charge people (including JSTOR) for access to electronic copies of the articles. Journal publishers are an anachronism that is being kept alive by an out-of-control copyright system; JSTOR is just an outgrowth of the problem, like a little hair growing out of a cancer (and make no mistake, "cancer" is an apt description of journal publishers: they get in the way of scientific research and make it harder for scientists to make their findings available to humanity).

All of this costs money. Universities and companies pay for JSTOR so that it can keep doing its job. More importantly, universities and companies pay for JSTOR so that they don't have to take on the burdens of maintaining complete archives of all academic works.