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 with the 5 Star Movement in Italy and Podemos in Spain, the Pirates have seized on voter distrust sparked by political scandals and economic missteps since the 2008 global financial crisis, when Iceland’s three main banks collapsed under mountains of debt.

The 5 Star Movement won mayoral elections in both Turin and Rome in June while Podemos has shown a bigger presence in recent general elections.

In their waterside headquarters in Reykjavik, decked out with pirate paraphernalia and which they call Tortuga, Pirate Party members talk of their rapid journey from a narrow internet-freedom-focused fringe to a force with views on issues including law and order and health care. In spring 2013, the Pirates cleared the 5% hurdle and entered Parliament with three lawmakers, including poet and co-founder Birgitta Jónsdóttir.

An offshoot of the Swedish internet-freedom agitators of the same name, which also spawned a similar movement in Germany, recent Pirate Party campaigning has centered on ideas for more direct democracy while vowing not to disrupt an economic recovery.

The party supports the draconian capital controls that were introduced to protect the national currency after the 2008 crash. The party has also backed recent government efforts to unwind fund-transfer restrictions.

“Our critics seek to portray us as upstart kids with too much time on our hands, but we are actually pretty sensible,” said Smari McCarthy, a party founder who has been suggested as finance minister in a Pirate Party-led administration.

The Pirates are seen taking 22.6% of the votes on Saturday, according to a survey conducted over the five days to Oct. 19 by the University of Iceland. That would top the 21.1% forecast for the center-right Independence Party, which has ruled for most of the roughly seven decades since the North Atlantic island of 330,000 people gained independence from Denmark.

Now half of a governing coalition with the centrist Progressive Party, Independence says its recent progress on the economy has earned it a renewed mandate.

Iceland is in the process of normalizing relations with creditors and the vital fishing and tourism sectors are both doing well. The economy is expected to grow about 5% this year, much faster than the European average, according to the central bank. Unemployment is seen at 3.3%.

But many voters worry their traditional leaders haven’t learned their lesson.

“We need something new,” said Benedikt Thorsteinsson, a 27-year-old waiter in a Reykjavik cafe. “The current government will only lead us back into crisis.”

Attitudes like that hark back to the 2008 crisis, which sparked widespread protests.

“There was a feeling that the political system had failed and there was a broad-based call for a new Iceland, which hasn’t yet been answered,” said Magnus Helgason, an economic historian at Iceland’s Bifrost University.

Recent political scandals have compounded the problem.

Documents leaked from the law firm Mossack Fonseca in April, known as the Panama Papers, showed Progressive Party leader and then-Prime Minister Sigmundur David Gunnlaugsson previously held an undeclared stake in an offshore company, which caused an outcry here.

Mr. Gunnlaugsson, who denies wrongdoing, resigned and was succeeded by an ally.

“Voting for the Pirates is an expression of protest against the establishment,” said Gunnar Helgi Kristinsson, a political scientist at the University of Iceland. “Their support reflects voters’ hopes that political renewal through replacing the established elites is possible and desirable.”

Whichever party comes in first on Saturday looks likely to need to form a coalition. Seven parties look likely to win seats. The Progressive Party is seen getting 9.1% of the votes, according to the University of Iceland survey.