Did a google search for "who must file 1099."
One link says that you only have to file if engaged in business or trade for profit, so maybe nonprofits don't have to file. But if the purpose is to prevent people from underreporting income, why exclude non-profits?
Second google search for "do nonprofits issue 1099." Please note that Google pre-populates when typing in a search query. I started typing "do nonprofits" and then Google provide the rest from a list of items to select. The beauty of Google is that if you have a question, hundreds if not thousands before you have asked the same question. One of the top hits is an IRS.gov page:
To quote: " However, nonprofit organizations are considered to be engaged in a trade or business and are subject to these reporting requirements. Other organizations subject to these reporting requirements include trusts of qualified pension or profit-sharing plans of employers, certain organizations exempt from tax under section 501(c) or (d), farmers' cooperatives that are exempt from tax under section 521, and widely held fixed investment trusts. Payments by federal, state, or local government agencies are also reportable."
In summary, yes, nonprofits must issue form 1099.