C corporations can not file as a non-profit. Nonprofits operate under section 501(c) of the Internal Revenue Code. C Corporations pay taxes under Chapter C of the IRS tax code, which is where the name comes from.
A not-for-profit corporation may be formed under the Not-for-Profit Corporation Law as a “charitable corporation” or a “non-charitable corporation” unless it may be formed under any other corporate law of New York State Ref: https://Certificate of Incorporation for Domestic Not-for-Profit Corporations.
You are correct in that the answer to your question is based upon a legal decision. The organization you are assisting should seek legal advice on how to structure their business. Once this has been done then the appropriate tax returns can be filed.
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