Yes. When you start a zip job, pass resultsEmail with the address to notify. When the archive is ready, S3zipper sends an email containing the download link.
You can also set the subject with emailSubject (the default is "Results from zipping"), which makes it easy to label results per project.
If you self-host the API, the email is sent through the SMTP settings in the API configuration (smtp_host, smtp_username, smtp_password, smtp_from), so you can use your own relay and branding. Without SMTP configured, the message is logged instead of sent.