Howdy Melinda McCaw,
That is not something that is currently possible with The Events Calendar. However we are planning to add a feature that is quite similar if not identical to this. It will allow users to subscribe to events, and for the admins to send out email blasts. There will be a lot of functionality centered around email and staying in touch with users, and it will likely be out before the end of the year.
I hope that info helps. Please let me know if that answer your questions or if you have any more. Cheers!
– Brook