The chat tab to the right of the UStream.tv video is not the best chat in the world. Scrolling is a pain for one thing and it doesn't offer any handy features that one expects in a good chat program.
But there's a better way. It turns out UStream is just using IRC (Internet Relay Chat) to implement the chat behind the scenes. This means that you can use a standard piece of software called an IRC client to connect. All you need are the right connection details.
Here they are:
Server port: 6667 (no SSL)
User name: your account name at ustream.tv
Password: your password at ustream.tv
Join room: pennssundayschool
And that's it! Learn how to use your IRC client and you should be able to do some tricks like filter the conversation, have it beep when certain words (like your name) are mentioned, and so on.