MOSCOW—One morning in September, Andrei Yegorov woke to discover his bank account was frozen and in the red to the tune of 75 million rubles, almost $1.2 million.

Across Russia, opposition activists say hundreds of people have found that they, too, inexplicably are 75 million rubles in the hole—the same amount state investigators allege Russian opposition leader Alexei Navalny’s organization had laundered for underworld crime lords.

Russia’s...