chock (n.)

c. 1400, chokkeful "crammed full," first element possibly from choke "cheek" (see cheek (n.)). Or it may be from Old French choquier "collide, crash, hit" (13c., Modern French choquer), which is probably from Germanic (compare Middle Dutch schokken; see shock (n.1)).

"jammed together," 1840, nautical, said of two blocks of tackle run so closely they touch; from chock + block (n.1) in the nautical sense "a pulley together with its framework."

"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.