I'd like to customize my chat box (Watson Assistant Chat Bot) and include my own CSS. I know I can set some changes on the plugin panel but I'd like to do more. Is there a way to do it?
Your best bet is to leverage WordPress itself to customize the look and feel via CSS.
You can do so from the sidebar menu Appearance -> Edit CSS.
Alternatively, your theme might offer you an option to write custom CSS code (and there also free custom CSS plugins you could install). But unless you have a good reason not to, the suggestion above should work well for you.