Tax-exempt organizations do not use Form 7004. They use Form 8868, which extends Form 990, 990-EZ, 990-PF, 990-T, and related returns by six months. For a calendar-year organization the return is due May 15 and the extension moves it to November 15.
The deadline
The 15th day of the fifth month after year end. Calendar year, May 15. A June 30 fiscal year, November 15.
Why this deadline is different
Missing a Form 990 deadline carries a daily penalty, and missing it three years in a row automatically revokes the organization's exempt status. Reinstatement means reapplying, paying the user fee, and explaining the gap to donors. No other extension on this site protects against something that severe.
Form 990-N
The smallest organizations, those with gross receipts normally under $50,000, file the electronic postcard, Form 990-N. There is no extension for it, and there is no penalty for filing it late, but the three-year revocation rule still counts a missed year.
Form 990-T
If the organization has unrelated business income, Form 990-T is a separate return with its own Form 8868. The extension does not extend the tax.
Common mistake
The volunteer treasurer files Form 7004 because it is the "business" form. Form 8868 is the only extension the IRS recognizes for 990-series returns.