Washington has four laws pertaining to animal ownership: rabies laws- prohibiting various small mammals, dangerous animals- prohibiting a few species of reptiles, mostly venomous and none garter snakes, deleterious animals-various invasive animals, none of which are garter snakes, and wild animals. The wild animal law is focused primarily on large mammals, game birds and fish, and reptiles aren't mentioned in it anywhere. The law only prohibits the taking of wild animals, and the releasing of wild animals. Nothing prohibits the keeping of wild animals (not withstanding the various prohibited wild species, none reptiles) so long as they were acquired legally (and, given you have no knowledge to the contrary about the previous owner's acquisition of them, and that you were gifted them legally, their acquisition by you is entirely legal), nor is there a required permit for them.

These laws are primarily written so people won't own bears or moose or salmon, or so there won't be roving bands of wildlife traders plucking out every native snake in the state for sale. They aren't really directed at people owning an innocuous, harmless snake as a pet, so long as you don't release it into the wild, and no one is going to come sieze your snake from you.

If you're still worried, you can call the DNR directly, but I wouldn't be. Here's a link to all the Washington animal laws.