Write into a database stored in App Group container from extension
Figured it out.
For keyboards, there is a property in the Info.plist file called RequestsOpenAccess. You can turn this on to enable write access to the shared container.
However, the end user will have to turn on "full access" in their settings for this keyboard, and they will be prompted with a scary message about how you could be logging their keystrokes.
Here's the Apple article that explains it: