Robert Rodriguez and Migdalia Wharton, a married couple in Orlando, Fla., have been out of work for more than a month and can’t afford to pay their credit-card bills.

When they called Capital One Financial Corp. to explain, the bank told them they could skip their April payments. But they doubt they will have money in May. Ms. Wharton, a school-bus driver, was told she wouldn’t get paid until school reopens. Mr. Rodriguez, a cancer survivor, is worried for his health and has stopped driving for Uber.

“We...