GANGNEUNG, South Korea—Last year, South Korea’s curling federation was embroiled in turmoil, lacking a leader, public support and even proper access to practice facilities.

Today? The South Korean women’s curling team has emerged as the breakout star of the Pyeongchang Olympics, an overachieving underdog that has burst into medal contention in a sport that South Korea had virtually no presence in just a few years ago.

South...