As per the title, is there a link anywhere in the mobile app that leads to the chatrooms? It's annoying to have to Google it every time on my phone. If not, it would be a useful feature.
No, there isn't. The iOS app offers this functionality via the 'More' tab:
but the Android app has a different navigation structure.
In your case, I'd probably just bookmark the URL in Chrome: https://chat.stackexchange.com/?tab=site&host=chess.stackexchange.com or even make it a shortcut on the 'desktop'.