The best fact about the woodchuck tongue twister is that its central pun does not exist. There is no wood in woodchuck and no chucking either. The word is a mishearing that hardened into English.
An Algonquian word, rebuilt
Woodchuck descends from an Algonquian term - forms such as the Cree otchek and the Ojibwe otchig are the ones usually cited by etymologists. Confusingly, that original word referred to the fisher, a dark forest-dwelling member of the weasel family that is not related to the woodchuck at all.
English speakers encountering an unfamiliar word did what English speakers reliably do: they reshaped it into something pronounceable out of parts they already knew. Otchek became woodchuck, and a plausible-looking compound was born.
This is called folk etymology
The process has a name. Folk etymology is what happens when speakers unconsciously remodel an opaque word into a familiar-looking one, and English is littered with the results.
The giveaway is always the same: the new form looks like it should mean something, and that meaning has nothing to do with the thing being named. Woodchuck is a textbook case, which is why it turns up in linguistics courses as often as in trivia columns.
Where groundhog comes from
Groundhog, by contrast, is straightforwardly descriptive and was coined in English. Early settlers saw a stocky, ground-dwelling animal with a considerable appetite and named it accordingly.
Both names are equally valid for the same species, Marmota monax. Neither is more correct than the other, though groundhog has the advantage of meaning roughly what it says.
What this does to the tongue twister
It makes it better. Read naively, the twister is a silly question about an animal's throwing capacity. Read with the etymology in hand, it is a joke resting on a mishearing that English made centuries ago and then forgot.
The animal was never named for chucking wood. We simply misheard a borrowed word, built a plausible compound out of the pieces, and then spent a hundred years asking the animal to live up to a name it never had.
Frequently asked
Why is a woodchuck called a woodchuck?
The name is an English reshaping of an Algonquian word - forms such as the Cree otchek and Ojibwe otchig are usually cited - which originally referred to the fisher, an unrelated weasel-family animal. English speakers rebuilt the unfamiliar word out of the familiar parts wood and chuck.
What is folk etymology?
Folk etymology is the process by which speakers unconsciously remodel an unfamiliar or opaque word into one that looks like familiar words. The result appears meaningful but its apparent meaning is unrelated to the word's actual origin. Woodchuck is a frequently cited example.
Is woodchuck or groundhog the correct name?
Both are correct and refer to the same species, Marmota monax. Groundhog is a descriptive name coined in English, while woodchuck is a reshaped Algonquian loanword. Neither has priority.