mid-15c., "the weighting of a fishing line," verbal noun from plumb (v.). Specific meaning "water and drainage pipes" is recorded by 1875, American English.
THE apparatus by which the water from a reservoir is carried about over a building and delivered at points convenient for use, is called by the general name of plumbing. The word "plumbing" means lead-work; and it is used to signify this water apparatus of a house because the pipes of which it largely consists are usually made of lead. [Edward Abbott, "Long Look House: A Book for Boys and Girls," Boston, 1877]
Alternative plumbery also is mid-15c. Slang meaning "a person's reproductive organs" attested by 1975.