Griff Neal set his alarm for 3 a.m. on Friday so he could be among the first to apply for a loan under a new federal program designed to help small businesses weather the coronavirus pandemic.

Bank of America Corp. turned him away that same morning. The reason: The auto-parts supplier he runs, Encap Technologies Inc., hadn’t borrowed from the bank before.

“I was literally floored,” said Mr. Neal, the company’s president. Encap, he said, has had checking and savings accounts with Bank of America for two decades, and the bank manages its employee retirement accounts.

Small-business owners have flooded banks with applications for the Paycheck Protection Program, a $350 billion chunk of Congress’s $2 trillion coronavirus stimulus package meant to help small companies and keep workers off the unemployment rolls. Many were quickly rebuffed because they don’t have the right kind of ties to the banks offering the loans.

Bank of America changed its stance Saturday after customers inundated it with complaints, dropping the requirement that applicants have a prior lending relationship with the bank. But the episode rattled small-business owners, who are worried that the pot of money will run out before they are able to dip into it. The funds are being distributed on a first-come, first-served basis and could run out within weeks unless Congress steps in. Wells Fargo & Co. said on Sunday interest from customers indicated it has already “reached its capacity.”