Independent tire retailer TBC Corp. is accused in a new lawsuit of failing to collect federally-required information from customers that could be crucial in the event of safety recalls.

Bruce Exum Jr. and Emilie Palmer say TBC and its subsidiary National Tire and Battery failed to give their customers tire registration forms. These forms provide tire manufacturers with the customers’ contact information, so they can be reached in case their tires need to be recalled.

The companies allegedly sold millions...