The overseer of the dominant U.S. women’s gymnastics program will step into a harsh congressional spotlight on Wednesday, still struggling with internal upheaval after a sexual-abuse scandal and facing existential threats from litigation and Capitol Hill.

Kerry Perry, the new chief executive of USA Gymnastics, is set to testify in Congress on Wednesday about the organization’s handling of the crisis that grew out of former team physician Larry Nassar’s sexual abuse of athletes. Perry, who took over late last year, will appear...