Interesting question...it actually made me go research the topic since etymology has always been a hobby of mine. Here's what I came up with:
1907, shortening of taximeter cab (introduced in London in March 1907), from taximeter "automatic meter to record the distance and fare" (1898), from Fr. taximètre, from Ger. Taxameter (1890), coined from M.L. taxa "tax, charge." An earlier Eng. form was taxameter (1894), used in horse-drawn cabs.
The verb is first recorded 1911, from earlier noun use as slang for "aircraft."http://www.etymonline.com/index.php?term=taxi
