Relying on the Fed to help midsize companies weather the coronavirus was never going to be a perfect solution. The central bank can help healthy businesses gain access to credit through its emergency lending powers, but it cannot simply give them cash to help them survive. The Fed chair, Jerome H. Powell, warned lawmakers of as much when they were designing the coronavirus relief package, and has reiterated since that the central bank has no ability to hand out money.

“I would stress that these are lending powers, not spending powers,” he said in a public appearance in April.

Yet Congress made the Fed the United States’ primary solution for thousands of companies. Lawmakers gave the Treasury Department $454 billion to support Fed lending programs in the late-March coronavirus relief package, specifying that economic policymakers should use some of that money to set up a lending program for midsize businesses, which, while big, are often still too small to raise money by selling stocks or bonds.

The Fed announced the details of that “Main Street” lending program on April 9, earmarking $75 billion to support up to $600 billion in bank loans for those companies. No start date has been given, though officials have indicated that the program could be up and running within a few weeks.

Companies in need of cash may be unable — or unwilling — to tap the Fed’s facility, leaving a giant section of corporate America vulnerable and with little help. About 19,000 American companies have 500 to 10,000 employees, and they employ 30.3 million workers, Census Bureau data shows.

Even if the San Francisco Goodwill branch is eligible for the Fed program once it gets up and running — the initial program design is not well suited for lending to nonprofits, though the Fed may tweak that — taking a Main Street loan would be a difficult business choice. The business is largely self-funded, using the money it makes on resold clothing and goods to pay its associates, hard-to-employ people for whom the program serves as job training. Borrowing money would leave it with debt that it might struggle to pay back.

“It puts us in a precarious situation,” Mr. Rogers said. The company is relying on savings and donations that will run out within four to six months, he estimates. “Really, fundamentally, we need help.”