Old English storc "stork," from Proto-Germanic *sturkaz (source also of Old Norse storkr, Swedish and Danish stork, Middle Dutch storc, Old High German storah, German Storch "stork"), from PIE root *ster- (1) "stiff." Perhaps so called with reference to the bird's stiff or rigid posture. But some connect the word to Greek torgos "vulture."
Old Church Slavonic struku, Russian sterkhu, Lithuanian starkus, Hungarian eszterag, Albanian sterkjok "stork" are said to be Germanic loan-words. The children's fable that babies are brought by storks (told by adults who aren't ready to go into the details) is in English by 1854, from German and Dutch nursery stories, no doubt from the notion that storks nesting on one's roof meant good luck, often in the form of family happiness.