chap (n.)

"peddler, itinerant tradesman," Middle English form of Old English ceapman "tradesman," from West Germanic compound *kaupman- (source also of Old High German choufman, German Kauffman, Middle Dutch and Dutch koopman), formed with equivalents of man (n.) + West Germanic *kaup- (source also of Old Saxon cop, Old Frisian kap "trade, purchase," Middle Dutch coop, Dutch koop "trade, market, bargain," kauf "trader," Old English ceap "barter, business; a purchase").

This is from Proto-Germanic *kaupōn- (source also of Danish kjøb "purchase, bargain," Old Norse kaup "bargain, pay;" compare also Old Church Slavonic kupiti "to buy," a Germanic loan-word), probably an early Germanic borrowing (Boutkan says 1c. C.E.) from Latin caupo (genitive cauponis) "petty tradesman, huckster, peddler," which is of unknown origin.

Compare cheap (adj.). In Middle English and later, chapman also could mean "a customer, purchaser." In a c. 1200 work the Devil is þe chapmon of helle as "the purchaser of souls." 

"low in price, that may be bought at small cost," c. 1500, ultimately from Old English noun ceap "traffic, a purchase," from ceapian (v.) "to trade, buy and sell," probably from early Germanic borrowings of Latin caupo "petty tradesman, huckster, peddler," cauponari "to haggle" (see chapman). Compare, from the same borrowing, German kaufen "to buy," Old Norse kaupa "to bargain, barter," Gothic kaupon "to traffic, trade."

The sense evolution is from the noun meaning "a barter, a purchase" to "a purchase as rated by the buyer," hence the adjectival meaning "inexpensive," the main modern sense, via Middle English phrases such as god chep "favorable bargain" (12c., a translation of French a bon marché).

The sense of "lightly esteemed, common" is from 1590s (compare similar evolution of Latin vilis). The meaning "low in price" was represented in Old English by undeor, literally "un-dear" (but deop ceap, literally "deep cheap," meant "high price").

The word also was used in Old English for "market" (as in ceapdæg "market day"), a sense surviving in place names Cheapside, East Cheap, etc. To do something on the cheap "with very little expense" is from 1859. Cheap shot originally was U.S. football jargon for a head-on tackle; extended sense "unfair hit" in politics, etc. is by 1968.

German billig "cheap" is from Middle Low German billik, originally "fair, just," with a sense evolution via billiger preis "fair price," etc.

"to cut with a quick blow," mid-14c., of uncertain origin, not found in Old English, perhaps from Old North French choper (Old French coper "to cut, cut off," 12c., Modern French couper), from Vulgar Latin *cuppare "to behead," from a root meaning "head," but influenced in Old French by couper "to strike" (see coup). There are similar words in continental Germanic (Dutch, German kappen "to chop, cut").

Related: Chopped; chopping. Chopping-block "block of wood on which anything (especially food) is laid to be chopped" is from 1703.