I'm sure I remember a mobile app that would connect you with a random user at the appointed time, chat-roulette style (although I think this was voice only, no video).

A bit of Googling has found several chat apps (e.g. Holla, Wakie) but none that specifically mention an alarm feature, but I'm sure I remember it.

It has obvious potential for trolling and abuse, but maybe a premium version, where you pay a small fee for the best 'knocker-ups' could be popular!