The line started forming outside the Social Security office in suburban Glendale, Ariz., not long after sunrise, dozens of retirees and people with disabilities, shuffling papers, some leaning on walkers, all anxious to know whether President Trump’s government overhaul had put their safety nets at risk.

When 9 a.m. came, an employee emerged from the building with fliers asking the crowd to come back — once they had scheduled an appointment.

“I’ve called for days!” one woman yelled.

“We came from a long ways away,” said another. Still another let everyone know they had been handed a load of bunk, though she used a more colorful term.

With the stock market in turmoil and the economy under threat, beneficiaries might see their monthly Social Security checks as predictable amid the chaos rippling out of Washington. After all, Mr. Trump has promised not to cut Social Security benefits for the 73 million Americans enrolled.

But that promise has not insulated the Depression-era program once deemed the third rail of American politics. Thousands of worried and frustrated recipients have thronged local field offices, asking why the phone lines are jammed, whether their local offices will be closed by Elon Musk’s team of software engineers and technology executives and whether they will lose their benefits.

Waves of buyouts and early retirements have hobbled the staff at many local offices, and recipients say it has become harder to use the agency’s website and phone systems, or even be seen in person.