Nearly half of Miss America’s board has quit or been forced to resign in the wake of the organization’s decision to eliminate the swimsuit competition from the contest, and 22 state pageant leaders are seeking to oust Chairwoman Gretchen Carlson over concerns about the pageant’s new direction.

Ms. Carlson, a former Miss America and an early leader in the #MeToo movement, announced on June 5 that Miss America was dropping the swimsuit and evening-gown competitions and replacing them with an extended onstage interview. She said...