Hi from Just Answer. The answer is pretty simple. Generally, you do not need to send them a 1099.
1099 reporting is intended to tell IRS about transactions, so they can anticipate tax reporting. Since most nonprofits are corporations, and corporations are not required to be issued 1099's for most transactions (a check payable to YMCA of Tulsa, Inc. can't be cashed by anyone other than the Tulsa Y, so IRS can track that pretty easily), a 1099 will not be required for the donated cash.
Thanks for asking at Just Answer. Ask any follow up you have, or leave positive feedback to close out your question. I'm PDtax.