bivalve mollusk, c. 1500 (in clam-shell), originally Scottish, apparently a particular use of Middle English clam "pincers, vice, clamp" (late 14c.), from Old English clamm "bond, fetter, grip, grasp," from Proto-Germanic *klam- "to press or squeeze together" (source also of Old High German klamma "cramp, fetter, constriction," German Klamm "a constriction"), possibly from a PIE *glem- or *glom- "contain, embrace" (see glebe).
If this is right then the original reference is to the shell. Clam-chowder attested from 1822. To be happy as a clam is from 1833, but the earliest uses do not elaborate on the notion behind it, unless it be self-containment.
"to dig for clams," 1630s, American English, from clam (n.). Clam up "be quiet" is 1916, American English, but clam was used in this sense as an interjection mid-14c.