Currently Teamup Calendar does not support email invites to events or RSVP functionality. This will come in a future version.
A work around, is to email the event page URL to the attendees of the event. They can then open the event via the URL and from there add the event to their personal calendar. Whenever the event page is accessed thereafter, they will always see the latest info. Read more about sharing an event as a web page.
Although not necessarily related, Teamup Calenders also feature a notification system.
Keywords: rsvp, invite, request, accept, reject, ignore, attendees, participant