The New York Giants hit four inside-the-park home runs at Braves Field on April 29, 1922, en route to a 15-4 victory over Boston. George Kelly collected a pair of inside-the-parkers, and Ross Youngs and Dave Bancroft contributed the other two scoring dashes around the bases. You can find the box score and play-by-play recap of the game here.
George Kelly in 1916 (baseballhalloffame.org)