With a rush of applicants overloading servers, the second round of the federal small business relief program was off to a rocky start Monday morning. But bankers serving Martha’s Vineyard said they are working diligently to secure funding for those that missed out on the first round of loans.

The initial $349 billlion authorized under the Paycheck Protection Program (PPP) dried up in just under two weeks on April 16. In Massachusetts, 46,937 loans were approved in the first round, according to the Small Business Administration, which guarantees the loans. Congress recently replenished the funds with a fresh $310 billion for a second round of loans.

Bert Talerman, co-president of Cape Cod 5, which has a full-service branch in Vineyard Haven, said the bank has already secured guaranteed loans for 800 small businesses since Monday. He said the average loan was less than $50,000.

Mr. Talerman said the majority of applications approved this week were held over from the first round. And he and other bankers said a combination of backlogged applications, competition from larger banks and technical issues is making it difficult for new, often smaller applicants to get approved.

“There were several hundred [applications] we had left in our system that did not secure funding in that first round,” he said. “Gearing up for the second round, most lenders already had that chunk of applications they had been working on and were ready to submit.”

Rockland Trust Bank has stopped accepting new applications in order to manage backlogged applications from the first round. Assistant vice president Emily McDonald said the bank received 5,000 applications in the first round and secured nearly 3,000 SBA guarantees. During this round, she said the bank must first prioritize the remaining 2,000 applicants instead of accepting new ones.

The backlogged applications could hurt smaller businesses, bankers said. Applications from sole proprietorships and independent contractors were only made available in the middle of the first round. Now, banks are fielding a significant increase in applications from these businesses, many of which will be pushed behind applications from the previous round.

“[Sole proprietorships] were held back from the early part of the first round,” Mr. Talerman said. “We are dealing with unprecedented demand . . . We are trying to take care of everybody who is applying with us.”

Amid the flood of backlogged applications, bankers have also reported technical difficulties getting the loans processed.

When gates re-opened on Monday, Martha’s Vineyard Savings Bank president James Anthony said the link between the bank and the SBA crashed almost immediately. He said 28 of the first 30 applications failed to go through to the SBA, and were bounced back.

“The system was continuously crashing,” Mr. Anthony said. “We constantly were just banging on the system trying to get applications through, to varying degrees of success.”

Mr. Anthony said the SBA intentionally capped the number of applications coming from each bank to stagger the rush. But even when the bank was authorized to submit applications, the volume apparently overwhelmed the server, he said.

“It is extremely difficult to navigate,” he said. “We have people in tears on the phone. If you miss one [application] you could be missing $10,000 in someone’s account that is very important to them. In some cases it could be an employer with a multimillion-dollar payroll.”

Rockland Trust and Cape Cod 5 reported similar problems. Mr. Talerman said a team of employees are working beyond midnight, when the SBA server is less burdened, in order to pump applications through.

“You have to learn to operate in the environment that you are in,” he said. “The technology issues certainly make the process more difficult.”

MV Savings Bank has 30 employees working around the clock to process applications. Mr. Anthony said the bank received more than 800 loans applications in the first round, 400 of which were approved and guaranteed by the SBA, tallying up to $38 million in relief for local businesses. He said the remaining 400 applications were either incomplete or withdrawn by the businesses that did not wish to go ahead with the loan.

“Everyone who applied with us in the first round, who had an application and went through with it, every single one was funded,” Mr. Anthony said.

He added that the bank has received over 500 more applications since Monday.

Meanwhile, the SBA has taken action to aid small community lenders and ensure the system is not overburdened. On Wednesday, between 4 and 11 p.m., the SBA only reviewed loan applications from banks with less than $1 billion in assets, according to the SBA website.

With assets of about $845 million, Martha’s Vineyard Bank was among those who qualified, and Mr. Anthony credited the SBA for looking out for smaller businesses in this round.

“The SBA is faced with a difficult challenge, and that is getting $660 billion out to small businesses through banks,” he said. “The fact that they are modulating accessibility . . . shows they are making every effort to get the money directly to Main street.”