Kit B. Asks: How much wood would a woodchuck chuck if a woodchuck could chuck wood?

I think we need to realize that this is a myth.  Just because they don't chuck wood, does not mean that they can't chuck wood.  You see, woodchucks, or Marmota monax, as I like to call them, are rodents, just like beavers.  Beavers chuck wood, so a marmota monax should be able to as well.

Your question is also lacking a clear definition of chuck.  It has several.  Seriously, you should look it up in a dictionary.  I think that the most obvious definition would be a short throw, however, it could also mean to get rid of, or also to vomit, such as "up chuck."

If you did indeed intend for it to mean vomit, then your question has merit.  Many rodents, such as mice and rats, lack the ability to vomit, the woodchuck may fall into this catagory as well, wikipedia did not mention it one way or the other.  But in this case, the answer is none.  You see, woodchucks don't eat wood.  They eat grass, berries, gardens, and sometimes insects; no wood.  Termites are about the only creature that can digest wood.  So, to summarize, woodchucks probably can't chuck period, and even if they did, they couldn't chuck what they wouldn't eat.  However, they probably have about 6 oz. stomach. 

On the other hand, if you intended for it to mean throw, there is much more variability.  A beaver for example does not really chuck wood, he drags it.  Now don't get me wrong, I can't chew down a tree with a 10 inch diameter overnight, but I could throw it farther than a beaver.  I mean throw the tree farther than the beaver could, not that I could throw the tree farther than a beaver.  Although, it may be hard to hold the beaver, I bet he would get kind of squirilly, beavery? as I heaved him around in circles over my head, and if he were to bite my hand and not let go...  The beaver will be our reference all the same.  A beaver can be 40 pounds, marmota monax, gets to be about 10.  You may think that this would mean that a woodchuck could chuck 1/4 of what a beaver does chuck.  But it is more complex.  Woodchucks burro in the ground, this gives them strong front limbs compared to the swimmer legs of the beaver, I bet they never have trouble getting the lids of jars of pickles or spaghetti sauce, but at the same time they have a highly curved back like a mole, which may make it difficult to "put their backs into it."  Woodchucks however tend to be aggressive, and you know how sometimes in an emergency you get like super human strength, well, I bet the woodchuck could harness that rage to chuck more wood then a beaver. 

On the other hand, woodchucks live in clearings, with little wood around them, and are rarely far from their holes in the ground.  This would make it difficult to find wood to chuck even if they had that insatiable desire deep in their cute little monax souls. 

So, on an wood chucking scale from 1 to beaver, we take beaver divided by 4 due to size, plus three for the strong upper body, minus 1 for the bad back, a 1.5 pissed off multiplier, last, minus 2 for no wood.  If I got the math right the answer is:

As much wood as a woodchuck could chuck if a woodchuck could chuck wood.

No beavers were harmed in the writing of this answer.  At least that I know of.