FRANKFURT—Volkswagen AG said Monday it would offer a $1,000 goodwill package to U.S. customers whose diesel vehicles are affected by the emissions scandal as a “first step” in the German car maker’s effort to rebuild trust among consumers.

As part of the package, Volkswagen is offering $500 to customers in the form of a prepaid Visa loyalty card. It is also offering a $500 Dealership Card that can be used at participating Volkswagen dealers.

The...