Here are a couple ways to find a copy of the articles of incorporation. The most direct method is to demand to see them. Under many states' nonprofit law, the organization is required to show them to you as a member. If there is a "principal office," they have to keep a copy at the office and make it open to inspection by members at all reasonable times during office hours. If the organization does not have an office, it is likely required to give a copy to a member upon request.
You also mentioned that the organization is a 501(c)(3). If that is the case, the organization is also required to make certain IRS documents available for public inspection. Those documents include the form that was submitted to get tax-exempt status (Form 1023) and the federal Form 990s that they have filed in prior years. If the Articles were attached to the From 1023, then they must allow the public to inspect those. But even if not, if you look at a copy of those IRS forms, they will include the federal tax ID number. Once you have the tax ID number, you can try searching the state's websites (Secretary of State and Attorney Generals). Some states require nonprofits to file copies of the Articles of Incorporation with the Attorney General, and make them available on the state's websites.