OED notes that German lampen "to hang limp" (Middle High German limphin) "has been compared." Perhaps it is from a PIE root meaning "slack, loose, to hang down" (source also of Sanskrit lambate "hangs down," Middle High German lampen "to hang down"). Related: Limped; limping. Limpen in Middle English was a different verb, "to happen, befall, fall to the lot of," from Old English limpan, which might ultimately be from the same root.
also limsy, 1825, a colloquial New England form of limp (adj.). For the formation, compare cutesy, drowsy, flimsy, tricksy, tipsy.
"a halting or limping, a limp," 1550s, from French claudication (13c.) or directly from Latin claudicationem (nominative claudicatio) "a limping," noun of action from past-participle stem of claudicare "to limp, be lame," from claudus "limping, halting, lame," which is of unknown origin. Related: Claudicant (adj.); claudicate.