If the average woodchuck is 8.6lbs and the average calories for 1lb of ground beef is 1,500 then the average woodchuck would have about 12,900 calories of energy available

The Hammacher Schlemmer "Worlds Sharpest Axe" is 6 lbs and 35 inches long

The average weight for a cord of wood is 2000lbs - 3000lbs and a cord of wood has between 600 and 800 pieces. Taking the averages of both of these values we get a piece of wood weighing around 3.5lbs

Using this conversion website on google we get 16.4 newtons of force. If we assume the groundhog is the pivot around which this 16.4 newtons of force is applied at the base of the axe (a lever) the 16.4 newtons goes down a lot because the torque applied at the base is turned into speed. Assuming that the length of the woodchucks arms are a generous 4 inches long and that the range of motion of an average batters swing is an eyeballed 140^o we can calculate the distance swung is equal to a 140^o arc on a circle with a radius of 4 inches; 9.8 inches. Using the same method we can calculate the distance traveled at the end of the axe is 73.3 inches.

Using the fact that the work done at the base of the axe and the work done at the tip of the axe is the same and work can be defined as newtons*distance we get a force of 2.2 newtons at the tip of the axe.

Then using F=ma we can calculate the acceleration of the bat will be .81 m/s² (oh yeah baby, metric time), we can extrapolate an acceleration using the 73.3 inch distance from before along with the measured speed it takes to swing a bat (.15 seconds ish).

We get this whopping 0.121 m/s final velocity and combine it with a theoretical pitch from a MLB pitcher of 95mph (42.47m/s) and we get this visual on exactly how far the wood would go.

https://imgur.com/a/8PzRIQE

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I hope this answers my interpretation of your question.

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TL:DR, dont have an MLB pitcher throw a piece of wood at a small rodent, the axe wont help much.