I embed the chat exactly as it states in the documentation, with an iframe:
https://github.com/justintv/Twitch-API/blob/master/embedding.md#chat
My website is hosted by Wix.com and their website editor. I do know that all of their user-customizable JavaScript code lives within an iframe running under the domain yourdomain.com.usrfiles.com
However, I specify just yourdomain.com in the Twitch redirect URI. Would this be the cause of the cross-domain errors? I did see in the Twitch chat JavaScript that it does window.parent in a few places.