RICH SQUARE, N.C.—The coronavirus hit rural Northampton County last month, but this has been the hardest week yet for Frank Timberlake, owner of Rich Square Market, the only grocery store in town.

Storms wiped out power and the market went dark for seven hours Monday, threatening to spoil the meat and milk he had stockpiled for customers. He sat in his truck trying to get someone on the phone at the power company.

“I...