WASHINGTON—Lawmakers at a recent House hearing on the future of the Export-Import Bank were given an extra piece of reading material: a personalized index card laying out exactly which companies in their districts benefit from the financing agency and how many people they employ.

The cards, which supporters of the bank plan to give to every member of Congress in coming weeks, are part of a lobbying push by corporations such as Boeing Co. and General Electric Co., and business groups such as the U.S. Chamber of Commerce and...