Five days after the start of a $349 billion emergency effort to get money into the hands of small businesses, the agency at the heart of the program is emerging as its biggest bottleneck.

The Small Business Administration, lightly staffed and working with aging technology, has been caught unprepared for the onrush of demand from desperate small-business owners who urgently need these loans as the coronavirus stalls the economy. In a boom year, the agency backs $30 billion of small-business loans — about the same amount that banks are now seeking on behalf of their customers in a day.

Larry Kudlow, the director of the National Economic Council, said on Tuesday that 178,000 loans totaling $50 billion had been approved for small businesses through the Paycheck Protection Program that was unveiled Friday by the S.B.A. and the Treasury Department. But bankers, small-business owners and others participating in the program say very little of that money has actually reached companies seeking the cash. The delays are causing confusion and panic among borrowers, especially those who see Trump administration officials playing up the program’s success. They worry they are being left behind.

“The expectation that this $2 trillion package would go through Congress and that the money would be flowing three days later, that was never a realistic expectation,” said Patrick Ryan, the chief executive of First Bank, a lender based in New Jersey. “But I get why people are frustrated.”