Due to a change of the connection mechanism with Apple, iCloud calendar and address book connection is only available with Doodle’s free app or via an ICS-Feed.
Why don’t you offer an iCloud connection on the web anymore?
Apple doesn’t support an official way to connect to users' iCloud accounts, except on the iPhone. In the past, Doodle used a 3rd party tool, which unfortunately does not work anymore due to a change in the connection mechanism with Apple. We want to provide the best possible experience, so we decided not to support the iCloud connection as long we have don't have a satisfying solution.
How can I sync my calendar with iCloud now?
You can download the free Doodle app and use iCloud from within the app.
How can I show my availability based on the iCloud calendar in MeetMe?
You can connect iCloud calendars via an ICS feed as described here.