By "support", do you mean "will not crash on encountering an unexpected verb"?
WEBDAV and CALDAV rely on a slew of new HTTP verbs; but IME only specialized servers actually support these verbs, in the sense of knowing what to do with them. There's an addon module for Apache that supports WEBDAV, but the logic to support CALDAV is insanely complicated, and nearly all implementations rely on a real database server.