By Bob Davis, Ruth Simon and Peter Rudegeair

The federal government's $350 billion small business loan program got off to a rocky start Friday, with some of the nation's biggest lenders saying they weren't yet able to process loan applications, discouraging business owners struggling to stay afloat.

"I've driven around the Los Angeles area for three hours and called several banks," said Paulo Amaral, the founder of Ally Right, a Los Angeles-based startup that helps make websites accessible to people with disabilities. "No luck at all."

By around 4:30 p.m., at least 9,779 loans worth about $3.2 billion had been approved, according to Small Business Administration chief Jovita Carranza, who was providing a running count on Twitter. Michael Strain, an economist at the American Enterprise Institute, calculated that half that amount would keep around 400,000 employed for the next eight weeks.

But with millions of people losing their jobs weekly, that is "a drop in the bucket," said Brock Blake, the chief executive of Lendio Inc., a technology firm that matches lenders with small-business borrowers.

Friday was the first day for the new Paycheck Protection Program, a part of the $2 trillion stimulus package designed to address the economic fallout of the novel coronavirus pandemic.

Businesses with 500 employees or fewer are eligible for the loans, which have a 1% interest rate and are designed to keep employees on the payrolls for eight weeks, limiting the mounting ranks of workers applying for unemployment checks. If a borrower doesn't lay off workers, the government plans to forgive the loan, including interest. Borrowers may also use the money for rent and utilities.

Many of the nation's biggest banks, including JPMorgan Chase & Co., Citigroup Inc. and Wells Fargo & Co., said Friday morning they weren't yet ready to take loan applications. By Friday afternoon Chase said borrowers with a business checking account could start applying for the loans online; but the bank warned of possible "processing delays and system failures" due to high volume and told customers that some applications wouldn't be approved by the SBA, given limited funds.

One reason many banks weren't ready to process applications was a delay in getting final rules about the program from the Treasury Department and SBA, as well as uncertainty about what kinds of documentation will be accepted to qualify prospective borrowers. The government didn't release final versions of the documents that must be filed by borrowers and lenders until Thursday night.

Bank of America Corp. started accepting applications Friday morning and had received more than 85,000 for loans totaling $22.2 billion as of 5 p.m., a spokeswoman said. But the bank said it is only considering customers with both outstanding loans and deposit accounts in order to speed up the approval process.

"First, we have to focus on the borrowing clients to make sure we can take care of them," Bank of America CEO Brian Moynihan said on CNBC. "And then secondly the people with the core relationship that don't borrow, we'll handle them also."

Sen. Marco Rubio, chairman of the Senate Small Business Committee, criticized the bank and said it should receive all comers. "This money is 100% guaranteed by fed govt," the Florida Republican said in a tweet.

Based on Ms. Carranza's figures, the average loan amount approved was about $325,000. At that rate, the roughly $350 billion allotted by Congress would run out after one million applications are approved. The government has said it approves loans on a first-come, first-served basis.

"If funds run low, there will be immense political pressure on Congress for funding to make sure that people aren't shut out because they are at the end of the queue," said Harris Simmons, CEO of Zions Bancorp, a big Utah lender.

Treasury Secretary Steven Mnuchin has said he would ask Congress for more money if the Paycheck Protection Program needs more funding.

About one-fourth of small businesses have already closed, according to a survey of 500 firms released Friday by the U.S. Chamber of Commerce and MetLife. Another 40% said it is likely they will have to close at least temporarily within the next two weeks.

Questions about the program remained even as it got under way.

One issue is whether the 1% interest rate is simple interest or compounded and what banks should do in case borrowers miss payments, said Scott Pearson, a financial services expert at the law firm of Manatt, Phelps & Phillips.

Also missing was a standard loan agreement that sets out the terms of the loan and conditions.

Chris Hurn, chief executive of Fountainhead Commercial Capital, a nonbank lender that specializes in SBA loans, said that as of Friday he was still awaiting guidance from the SBA on several issues, including how to submit the forms that both applicants and lenders are required to fill. Mr. Hurn said he couldn't begin processing applications without such guidance.

The SBA didn't respond to a request for comments on the complaints.

Other small-business owners say they are frustrated by policies, such as the one outlined by Bank of America, that put certain customers ahead of others. Giving those with existing lending relationships priority penalizes small businesses that have taken a prudent approach to debt, but are still suffering financially from the pandemic.

Jen Storey, co-owner of Redhill Painting in San Francisco, has a business checking account with Bank of America and a business credit card with JPMorgan Chase. She said the eight-person company that she and her husband have run since 2004 doesn't meet the standards for processing the new SBA loans set by either bank.

Mr. Amaral of Ally Right said he rose before sunrise to get an early start applying for a loan at a local SBA lender listed on the agency's website. The lender closest to his home said it no longer makes SBA loans, the second didn't know anything about the new program and the third was closed because of the Covid-19 virus. He also tried Wells Fargo, where he has business and personal banking accounts, but said bank employees didn't know about the program.

A Wells Fargo spokeswoman said the bank would only accept applications for the loans online and that it has signs in its branches informing customers of that policy.

Some banks were working hard to update their systems to deal with what they expected to be an avalanche of loan applications. Boston-based Eastern Bank won't begin processing applications until next week because it needs to update its technology, according to Quincy Miller, the bank's president and vice chair.

Mr. Miller said guidance came too late from federal agencies to make all the necessary changes in time to start processing the loans Friday. Once the bank gears up, it will give priority to applications from customers that have any existing relationship with Eastern Bank.

Live Oak Bancshares, Inc., the largest lender in an SBA flagship program, said it is working through a pipeline of over 1,000 loan applications from its roughly 4,000 existing customers. Then, it plans to open up the program to businesses near its North Carolina headquarters followed by new customers in the big industries it serves, which include veterinary offices and chicken farms

The bank submitted its first paycheck protection loan application to the administration at 4 a.m. Friday, said Huntley Garriot, president of the Wilmington, N.C.-based bank. It took more than one try.

By Friday afternoon, the bank received its first loan authorizations from the SBA, and it hoped to get the money out to customers either by the end of the day or Monday.

Mr. Simmons, the Zions CEO, said his firm will be working through the weekend to catch up with the applicants that want to file loans. He said the bank is training 3,000 of its 10,000 employees to handle the paycheck loans. "We want people who can answer questions" from borrowers, he said.

Becky Feinberg-Galvez, chief executive at Chicago-based Shop4Ties, a custom apparel company, had hoped to apply on Friday for a loan through Citibank, but received an email saying the bank wasn't yet prepared to accept applications.

"I don't blame [Citibank] obviously. I just think it's frustrating there's no finality and nobody can give you an answer," Ms. Feinberg-Galvez said.

--Amara Omeokwe and Yuka Hayashi contributed to this article.

Write to Bob Davis at bob.davis@wsj.com, Ruth Simon at ruth.simon@wsj.com and Peter Rudegeair at Peter.Rudegeair@wsj.com