In the United States, as a general rule, pretty much any adult resident can start an organization, some jurisdictions require more paperwork and fees than others. To start something along the lines of what is envisioned in this thread is for another discussion all together, and should take place with qualified professionals.
Once the organization and administrative matter has been resolved, the subject then turns to some of the issues that Fnord pointed out above, particularly fundraising. Contrary to the picture generated by some organizations and the media, most non-profits (in the broadest sense of the term) run on fairly limited revenues, and either have a limited number of donors, or spend a large amount of time fundraising. For some organizations, the fundraising is part of the outreach and recruitment effort.
I don't say the above to be negative, only to point out what would need to be resolved at the concept stage, before any documents are drafted.
I think part of Feralucce's comment is accurate in regards to non-profit organizations, in that staffing with the right mix of qualifications, availability, and interest would be difficult. It is not impossible, both ASAN and GRASP have shown it to be possible.
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