JAKARTA, Indonesia—Indonesians turned out by the millions across the island-nation Wednesday to vote for local leaders, who have an increasingly large role in driving projects and programs central to reviving Southeast Asia’s largest economy.

Final results won’t be announced until Dec. 18 and Dec. 19, but quick counts showed some early wins, with the popular incumbent mayor of Surabaya, Tri Rismaharini, grabbing a landslide share of the vote.

The...