SEOUL—South Korean legislators voted to impeach President Park Geun-hye following months of uproar over influence-peddling accusations, marking the downfall of the country’s most powerful political dynasty.

The resounding vote to impeach President Park is likely to ramp up uncertainty in the region. The final vote count was 234-56, more than enough to secure the two-thirds majority required for impeachment in the 300-member National Assembly.

Many...