I've been trying to find out of its possible to access Apple's iCloud calendar from a web application.
Does it support CalDAV or any sort of web API? I can't seem to find a clear answer for this.
iCloud uses CalDAV. A quick Google search returns this hint for determining server URL info for use with iCal on Mac OS 10.6 -- but once you have that info you should be able to use it with any CalDAV client.