What does chuck mean in in the tongue twister if a woodchuck could chuck wood?

they really just mean "cut"
To chuck is to throw. It is a verb, with a present and past tense. It can also be used as a metaphor, not literally, for "throw (out)".

I tried to chuck the wad of paper but missed the wastebasket.

He decided to chuck what he wrote and start over. --- Here, he is not actually throwing anything, but the word chuck is used as a metaphor for throwing (something out or away from him).

I wanted to chuck the dress I started to sew, but I didn't have time to make a different dress.

The man chucked the cut logs onto the pile. His wife sorted the chucked logs and stacked them into neat rows against the house.