I've emailed Automattic about this, and they gave me this answer:
Grunion itself doesn't have a CSV export, but there are several
WordPress CSV export plugins available. Since Grunion stores messages
as a standard WordPress custom post type, it's likely they'd be
exported by those plugins (though I haven't tried myself). If not it
would probably be easy to adapt one of them to do the job.
I'm confused as to what a "standard wordpress custom post type" is.
How do I go about exporting the information to a CSV file?
I've contacted Automattic later, but haven't gotten a response.