If it's a website Java applet, you're limited to whatever the applet gives you. You could try pressing Ctrl-A to select all and then Ctrl-C to copy, but that may or may not work (and may only save a recent bit of the chat).

If you can find out what IRC server and channel name the applet is using as a back-end, you can potentially connect to the channel with a real IRC client and get more functionality.