While a vague answer to this question is given in the second verse of the Mother Goose nursery rhyme ("as much wood as a woodchuck would chuck if a woodchuck could chuck wood"), a more precise answer — according to a 1995 study — is that woodchucks can chuck 361.9237001 cubic centimeters of wood per day. According to the Scientopia blog, a study focused on answering this question was published in the magazine "The Annals of Improbable Research" in 1995. Groundhogs, which, according to National Geographic, are also known as woodchucks, were used as the subjects of this study.
Groundhogs feed on grasses rather than wood, and they aren't known to throw or otherwise "chuck" wood in the wild. In the study mentioned by Scientopia, two researchers decided that "chuck" could mean "chew" in this context. Each groundhog collected for the study was given a 2" x 4" piece of wood and observed; the result was the finding that a woodchuck can chuck (chew) . This study was largely intended to be humorous, though certain research conventions were observed. Still, for those who desperately need an answer to the question, it's probably best to stick with Mother Goose's cleverly vague answer.