It’s pretty easy to make a chat type system in Bubble because of the neat auto-update the UI when the database changes built-in capability of Bubble. A Message table with a sender/recipient type of design.
This is a good place to start https://bubble.io/blog/build-whatsapp-clone-no-code/
And this plugin is useful for formatting
http://forum.bubble.io/t/chat-bubble-plugin-free-useful-plugin-for-styling-chat/101080