exchange song files

Is there a way for users to send Apple song files to another user on the same account?

If you're referring to Apple Music links, Admins and Editors can curate those in Planning: https://www.worshiptools.com/en-us/docs/78-editing-songs (read the Media section)
If you're referring to .mp3s, you could upload them as .mp3s in the same place as above or upload them as service attachments. Keep in mind that these uploads will take up space in your Cloud Storage.