REYKJAVIK, Iceland—Populist movements featuring unconventional politicians have been gaining footholds across a reeling Europe. In Iceland, the Pirate Party—co-founded by a poet and sporting a black Viking sail as a logo—now has a chance for power.

The Pirates, founded in 2012 and advocates of more direct democracy and greater internet freedom, are projected to win Saturday’s general election and get a shot at leading Iceland’s next government.

As...