On July 27, 2015, USA Gymnastics dispatched its future superstar, Simone Biles, on a round of promotional appearances in the federation’s hometown of Indianapolis. It was part of a hectic marketing tour into which then-chief executive Steve Penny also squeezed an appearance by Biles at a party for his daughter’s gymnastics group the night before.

That same day, Penny did something he did not tell Biles or her family about. He had USA Gymnastics contact the Federal Bureau of Investigation in Indianapolis to report concerns...