"to throw," 1590s, variant of chock "give a blow under the chin" (1580s), possibly from French choquer "to shock, strike against," imitative (see shock (n.1)). Meaning "pat playfully, give ablow to" is from 1610s. Related: Chucked; chucking.
"piece of wood," 1670s; "piece of meat," 1723; probably a variant of chock (n.) "block." "Chock and chuck appear to have been originally variants of the same word, which are now somewhat differentiated" [OED].
Chock and Chuck, Are low terms, very frequently used before full,—as the coach was chock full of passengers. The house was chuck full. [Daniel Powers, "A Grammar on an Entirely New System," West Brookfield, 1845]
Specifically of shoulder meat from early 18c. (the exact cut varies from place to place). Meaning "device for holding work in a lathe or other machine" is from 1703 (also chock). American English chuck wagon (1880) is from a mid-19c. meaning "food, grub," generalized from the meat sense.
"slight blow under the chin," 1610s, from chuck (v.1). Meaning "a toss, a throw" is from 1862.