Chap
late Middle English: of unknown origin.
wiktionary
Shortened from chapman(“dealer, customer”) in 16th century English.
From Middle English chappen(“to split open, burst, chap”), of uncertain origin. Compare Middle English choppen(“to chop”), Dutch kappen(“to cut, chop, hack”). Perhaps related to chip.
From Northern English chafts(“jaws”). Compare also Middle English cheppe(“one side of the jaw, chap”).
Shortening
etymonline
chap (n.)
1570s, "customer," short for obsolete chapman in its secondary sense "purchaser, trader" (also see cheap). Colloquial familiar sense of "lad, fellow, man or boy" is first attested 1716, usually with a qualifying adjective. Compare slang (tough) customer and German Kunde "customer, purchaser," colloquially "fellow."
chap (v.)
"to crack open in fissures," mid-15c., chappen (intransitive) "to split, burst open in fissures;" "cause to split or crack" (transitive); perhaps a variant of choppen (see chop (v.), and compare strap/strop), or related to Middle Dutch kappen "to chop, cut," Danish kappe, Swedish kappa "to cut."
Usually in reference to the effects of extreme cold followed by heat on exposed body parts. Related: Chapped; chapping. The noun meaning "fissure in the skin" is from late 14c.