BERLIN—Just months ago, the AfD was on a roll. With all-time best poll ratings, the German far-right party appeared headed for record election scores this year, promising to extend a winning streak for nationalist and populist parties across the region. Now, an avalanche of scandals is clouding the party’s national and European ambitions. After years of thriving on controversy, the AfD is seeing voters pull away and risks being relegated to a purely regional force concentrated in Germany’s former communist east. Speaking in her sixth-floor office overlooking Berlin’s monumental Reichstag building, Alice Weidel, the AfD’s co-chairwoman, acknowledged the difficulties. “Like...