You can download a transcript of the chat log from a past webinar within seconds using Demio's past event reporting.
First, click on Events from the top menu, then click on the webinar title you are needing the Chat analytics for
Then click on Insights for the session
Click the Chat tab to see all of the chat messages from that session. Then click Export
To export the chat messages from your attendees in a CSV, you can select Attendees & Chat messages