Old English wir "metal drawn out into a fine thread," from Proto-Germanic *wira- (source also of Old Norse viravirka "filigree work," Swedish vira "to twist," Old High German wiara "fine gold work"), from PIE root *wei- "to turn, twist, plait."
A wire as marking the finish line of a racecourse is attested from 1883; hence the figurative down to the wire. Wire-puller in the political sense is by 1842, American English, on the image of pulling the wires that work a puppet; the image itself in politics is older:
The ministerial majority being thus reduced to five in a house of five hundred and eighty-three, Lord John Russell and Lord Melbourne respectively announce the breaking up of the administration, and the curtain falls on the first act of the political farce, to the infinite annoyance and surprise of the prime wire-puller in the puppet-show. [British and Foreign Review, vol. IX, July-October 1839]
I looked in at the Alhambra the other night, and found an excellent show, notably, a high-wire act by Mdlle. Virginia Aragon. A very handsome Spaniard with coal-black tresses, she does her work with great neatness. The best thing she does is to kneel on the wire, and, leaning forward, pick up with her teeth from between her knees, a handkerchief. Then she swings on the wire, balancing herself with one foot only. Altogether, she is the smartest wire-walker I've seen for many a day. Her sister, by the way, is a trapezist and figured at the Empire not long ago. [The Sketch, Nov. 27, 1895]