How does one go about making a Storm pet? Is it like an specific event thing like with Galactic pets?

There is a Storm Potion. You can get it (a random part of a random event) from Tempest, or buy them from other users.

Adding to , you can also enter the Potion Lottery in hopes of getting a storm potion, or you can try zapping your pet. Occasionally, you can find Storm pets in Adopt R Us or being adopted out on the Adoptions sub-forum.

However, if you are after a specific Storm pet, I highly recommend just buying the potion from other users. Good luck~ ^^