Negotiations with management broke down over pay and other benefits.
The development comes a year after workers at the South Korean carmakers walked out for four months, costing Hyundai $1.5bn (£970m) in lost production.
Strike action by the 45,000 workers the union represents could begin as early as Tuesday next week.
South Korea’s car industry is subject to recurring strikes because of strong unions.
“We don’t like to strike. But the company has accepted part of our demand only after we staged strikes,” union spokesman Kwon Oh-il told Reuters.
The demands of the union include bonuses equivalent to eight months of base salary, as well as a 56.25g gold medal for those with 40 years’ service.
They are also asking for 10 million won for each workers whose children opt not to attend college. The company already pays tuition for college-going children of employees.
The vote in favour of industrial action was announced on Wednesday. However, the companies are engaged in a six-day mediation period. That means the earliest a strike could start is Tuesday of next week