Can a tax-exempt organization file one Form 8868 to extend multiple annual returns?

No. For each return, you need an extension; you must file a separate Form 8868 extension.

If your organization must file more than one type of annual return (such as both Form 990 and Form 990-T), you’ll need to file a separate extension request for each. Form 8868 only extends the due date for the specific return it is filed for.