The 19th-century expression comes from a toad eater. Yes, one who eats toads! While frogs have long been considered a gastronomic delicacy, all toads were thought to be poisonous at one time. But they were useful in 17th-century medicine shows that featured an assistant who'd eat one (or pretend to) so that his swindling master could demonstrate the curative powers of his elixirs. Toad-eater became toady, which now means a sycophant who does distasteful things to gain favor.
The New England woodchuck, or Marmota monax, has no connection with wood whatsoever, despite the popular jingle. Its name derives from the Cree Indian word wuchuk or otchcok, a fisher, which North American settlers corrupted to woodchuck and applied mistakenly to the groundhog.
SOURCES: 'Thereby Hangs a Tale,' by Charles Funk; 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson; 'Why You Say It,' by Webb Garrison; 'Who Put the Butter in Butterfly?' by David Feldman.