I have collected 50+ puzzle rings so far and I don't feel like running all these vaults manually. I know there was a plugin created that allowed vault runs when it was spawned via goblin so this shouldn't be too difficult to combine.
It already exists. It is one of the trinity profiles: Trinity->Profiles->ExploreGoblinRealm. It will cube puzzle rings and run the vault. Make sure QuestTools is up-to-date and enabled, and make sure your Trinity settings destroy destructibles if you want that.