What invitation system is in place for members to use for potential sign ups, if any? If not, is there a way we can leverage a system in place to be able to invite others to join the network?
Yes, it is set the same way it showed in the depiction you shared (via the links) however I just saw this error:
Notice
Could not start mail function.
Message
User saved.
This was shown when I was trying to create an admin user for the stackideas to be able to have access.
It might be when you access your site/easysocial, the cronjobs kicks in and thus showing you those errors. Can you consult with your current web hosting service provider regarding this issues and see how it goes.