Is there a way to turn off an email notification on a site basis?

You can turn off the notification on the site by removing all the recipients from who can receive the notification.
  1. Go to Sites > [site name] > Email Notifications.
  2. If the notification is the default, you'll have to click on the edit icon to view and create a custom notification.
  3. If the notification is already a custom notification, you'll need to edit it by clicking the 'pencil' icon.
  4. Any user listed under the 'Who receives this notification' will need to be removed.
  5. You can do this by clicking on the double left arrows in this area.
  6. All users will be moved to the Users list on the left.
  7. Click 'Save Changes'.