Kite posted a 6-under 65 in round two, while first-round leader O'Meara shot a 68, as the two share a two-stroke lead at 11-under-par 131.
Mark James held the lead during the second round, but a late bogey cost the former European Ryder Cup captain. He finished with a 5-under 66 and shares third place with John Cook (68), playing in only his second Champions Tour event.
Kite, a former U.S. Open champion, seeks his first victory of the year and his 10th on the Champions Tour.
O'Meara, who had 63 Friday, won the Masters and British Open in 1998. Since joining the senior circuit, O'Meara has not won, but has three runner-up finishes.