So, what are the Kami pets, and how/when can I get one?

Kami pets are special "deity" pets that can only be acquired during special "Kami events". One or more Kami pets may be available during the event (for example, the Gate to the Dreamlands Kami event last year had four Kamis you could earn, each with different prerequisites)

The most recent Kami event was Cure the Love around Valentine's Day.

Kami events consist of a series of tasks (pet quests, item collection, item crafting, etc.) that lead up to the ability to challenge the Kami in the arena. If you win, you get the Kami as part of your collection.

Kami events are announced in the news, but there have been no announcements for a new one since Valentine's Day.