James Foulis won his only major title, three strokes ahead of runner-up Horace Rawlins, the defending champion.
Shippen, a caddie at Shinnecock Hills, shot an opening round of 78, which placed him just two back of leader Joe Lloyd.
He was in a position to win the championship until an eleven on the 13th hole of the final round, and finished tied for sixth.
Foulis, the third-place finisher in the inaugural U.S. Open the year before, recorded rounds of 78-74 to prevail by three over Rawlins.
At 4,423 yards (4,044 m), Shinnecock Hills played as the shortest course in U.S. Open history.