It appears that this isn't available, even though the "Data Sources & Access" screen for it makes it sound possible. Even just read-only access would be super useful.
It's a bit frustrating that this and the medications data hasn't been opened up to third-party developers. I understand that the medication data is structured in a completely different way to the rest of HealthKit, so there's definitely much more complexity there, but the mood tracking records appear to be simple flat records like everything else.
I've filed FB13069949 about accessing State of Mind data.