Capturing live website Chats as Interesting Moments

First, to use the Interesting Moment activity type, you need to have Marketo Sales Insights active (you don't need more than 1 license, but you need to have MSI).

You can however use a Munchkin activity to represent the same thing, like a hit to the pseudo-web-page example.com/chat/engaged.  Or you could submit a Marketo form that reflects their engagement (in fact, if you don't do something like this, their Munchkin session remains anonymous, which is surely not what you want if you know who they are via other means).

The reason you don't see any Marketo documentation on the subject is that it's actually a question of the chat software's API. If it can trigger on events ("new chat", "chat closed", etc.) in JavaScript (preferrered) then you wire that up to Munchkin. If it has a back-end webhook functionality, have that send to an API gateway that can update fields, create Custom Activities or Custom Objects, or whatever you want in Marketo. But there's no one way to plug in a generic "chat" concept.