Transitive meaning "move (something) back and forth or up and down" is from c. 1300; of dogs and their tails from mid-15c.: "and whanne they [hounds] see the hure maystre they wol make him cheere and wagge hur tayles upon him." [Edward, Duke of York, "The Master of Game," 1456]. Related: Wagged; wagging. Wag-at-the-wall (1825) was an old name for a hanging clock with pendulum and weights exposed.
c. 1500, kind of small bird that has its tail in continuous motion (late 12c. as a surname), earlier wagstart (mid-15c.), from wag (v.) + tail (n.). From 18c. as "a harlot," but this sense seems to be implied much earlier:
If therefore thou make not thy mistress a goldfinch, thou mayst chance to find her a wagtaile. [Lyly, "Midas," 1592]
Proto-Indo-European root meaning "to go, move, transport in a vehicle."
The root wegh-, "to convey, especially by wheeled vehicle," is found in virtually every branch of Indo-European, including now Anatolian. The root, as well as other widely represented roots such as aks- and nobh-, attests to the presence of the wheel — and vehicles using it — at the time Proto-Indo-European was spoken. [Watkins, p. 96]
It forms all or part of: always; away; convection; convey; convex; convoy; deviate; devious; envoy; evection; earwig; foy; graywacke; impervious; invective; inveigh; invoice; Norway; obviate; obvious; ochlocracy; ogee; pervious; previous; provection; quadrivium; thalweg; trivia; trivial; trivium; vector; vehemence; vehement; vehicle; vex; via; viaduct; viatic; viaticum; vogue; voyage; wacke; wag; waggish; wagon; wain; wall-eyed; wave (n.); way; wee; weigh; weight; wey; wiggle.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit vahati "carries, conveys," vahitram, vahanam "vessel, ship;" Avestan vazaiti "he leads, draws;" Greek okhos "carriage, chariot;" Latin vehere "to carry, convey," vehiculum "carriage, chariot;" Old Church Slavonic vesti "to carry, convey," vozŭ "carriage, chariot;" Russian povozka "small sled;" Lithuanian vežu, vežti "to carry, convey," važis "a small sled;" Old Irish fecht "campaign, journey," fen "carriage, cart;" Welsh gwain "carriage, cart;" Old English wegan "to carry;" Old Norse vegr, Old High German weg "way;" Middle Dutch wagen "wagon."