The short answer is no. Not rarely, not occasionally, not under special circumstances. No woodchuck has ever chucked wood, and the tongue twister is built entirely on a coincidence of vocabulary.
What a woodchuck actually does
A woodchuck is a grazer and a digger. Its day divides fairly neatly between eating soft green plants above ground and maintaining an elaborate tunnel system below it.
Both activities are conducted close to a burrow entrance. A woodchuck feeding in the open is a woodchuck within sprinting distance of a hole, and it will pause repeatedly to sit upright and scan for anything that might eat it. Timber handling does not feature at any point.
But it does gnaw wood
Here is where the confusion has room to grow. Woodchucks are rodents, and rodent incisors never stop growing. Left unworn they would eventually make eating impossible, so every rodent must grind them down against something hard.
Woodchucks oblige by gnawing on wood - tree bark, roots, fence posts, the occasional deck support. So there is a real and regular interaction between woodchuck and wood. It is simply not throwing.
Gnawing is maintenance. Chucking would be freight. The two have nothing to do with each other.
Could a woodchuck chuck wood if it wanted to?
Physically, a little. Woodchucks have strong forelimbs and heavy claws built for excavating packed earth, and an animal that can shift hundreds of pounds of soil is not weak.
But the anatomy is wrong for the job. Those forelimbs are built to pull material backwards beneath the body, not to lift, grip and release something forwards. A woodchuck can move earth in enormous quantities precisely because it never has to pick any of it up.
Why the name misleads everyone
Because woodchuck looks like a compound of two ordinary English words that together describe an action. It is not. The name is an English reshaping of an Algonquian word for a completely different animal, and the resemblance to wood and chuck is an accident of pronunciation.
The tongue twister is therefore a joke about a mishearing - which is a far better joke than the arithmetic that usually gets attached to it.
Frequently asked
Do woodchucks chuck wood?
No. Woodchucks are herbivores that graze on green plants and dig burrows in soil. They have never been observed picking up or throwing wood, and their forelimbs are adapted for pulling earth backwards rather than lifting and releasing objects.
Why do woodchucks chew on wood?
Like all rodents, woodchucks have incisors that grow continuously and must be worn down. Gnawing on bark, roots and timber keeps those teeth at a workable length. Chewing wood is dental maintenance, not chucking.
What do woodchucks do all day?
They alternate between grazing on soft green vegetation near a burrow entrance and excavating or maintaining their tunnel systems. They are most active shortly after dawn and again before dusk, and they rarely stray far from an escape hole.