Well you should ask the people that made Omegle about an API. From what I've seen, it doesn't have an open API, but maybe they can give you some extra info.
You can also try Firebug. It lets you see all the requests that are made and their target. The site is AJAX driven, that's why you can't see the desination scripts directly in the page.
You can (1) build a program that sends post requests to the targets show in the Firebug net panel and passes the data to the chat-bot or (2) build a browser plugin that does that (it's pretty easy to do in Firefox). You may need the browser plugin because of cross-domain request limitations and because you may not see the received messages from an external program/web-app - it's not very clear to me how they implemented that, i guess it's something Comet-like.
Hope this helps :).