Three family-owned dealerships sued Volkswagen AG, alleging the German auto maker intentionally defrauded them when installing software on diesel-powered vehicles capable of duping government emissions tests.

The proposed class-action suit, filed in an Illinois federal court Wednesday, accuses Volkswagen of luring dealerships into investing in showrooms to sell environmentally friendly vehicles while the German company was secretly deceiving regulators in the U.S. It seeks recompense for losses dealers suffered from falling...