They first appear at the very beginning of the story, in Wario's stage. One Splunk sneaks into Wario's house while he is eating a pile of snacks, and steals them. Wario angrily chases him through Diamond City until they reach the Temple of Form, where Wario finds the Form Baton. A large boulder crashes down and he runs away. Once he escapes the temple, he decides the Baton will be a part of his next money-making scheme.
At the end of Tiny Wario's stage (as well as the story), a group of Splunks confront Wario, wanting the Form Baton back. Wario flees to the Temple of Form, where he trips and the Form Baton falls back into its place. A Splunk standing before him simply says "Thanks!", and the story ends.
They also appear in the ending of Orbulon's stage, who finds the Form Baton in the Temple of Form after his spaceship crashes. He tries to rule the Temple as his new spaceship, before the Splunks appear, take the Form Baton (and the Balance Stone) back, and throw him out.
Not much information about the species itself is given in-game, except for showing that they used to live with the ancient people of Diamond City. The Japanese WarioWare: Smooth Moves website dedicates a section to the Splunks on its "Introduction to the Temple of Form" page, largely consisting of uncertain data and speculation:
Names in other languages