Hey, I once used FullCalendar for a web-app. It is MIT-licensed and quite easy to use and supports older browsers thanks to jQuery. However, it depends on jQuery and Moment.JS, so it might be a bit bloated.
For my web-app, I used FullCalendar as front-end, added a few forms for creating new events (just handle onClick events) and put a login with multiple users to lock it down. The login is session-based and handled by a custom backend, which can be queried by FullCalendar. The API is well documented and there should be no problem implementing anything you need.