Desperate and defeated, Naqibullah Laghmanai was calling everyone he could think of Monday morning in Houston, trying to get his family out of Afghanistan. He was on hold with U.S. Citizenship and Immigration Services for at least the 10th time in recent weeks, he said. He was crafting a plea to the Canadian Embassy, though his family never worked for the Canadians. He was calling members of Congress.

On Monday, after the Taliban took Kabul, Taliban fighters searched his aunt and uncle’s home, looking for the men of the family, his mother told him.

“Their only hope is on me to get them out and I can’t,” Mr. Laghmanai, 31, said of his parents and siblings.

Mr. Laghmanai, who worked as an interpreter for U.S. armed forces starting in 2007 in Kandahar, joins other Afghans currently living in America who have spent the past 48 hours feverishly trying to save their family members who are left behind.

On Sunday, the Taliban nearly completed its takeover of the country when it conquered Kabul, the country’s capital. The U.S. military took control of the Hamid Karzai International airport Sunday and suspended all flights until security could be restored, officials said. Desperate Afghans have converged on the airport trying to find a way to escape.

Suneeta, an Afghan immigrant living in Albany, N.Y., has been trying to reunite with her four children, now ages 7 to 17, who are on their own in the Kabul area.

Mr. Laghmanai’s family in Kabul had already received a threat from the Taliban and had been in hiding for months because, he suspected, of his work for the U.S. government and his father’s work for the Afghan government. Special Immigrant Visas for his parents and young siblings have been pending for 15 months, Mr. Laghmanai said.

Recipients of such visas, which are granted to people who assisted the U.S. military, fear their family members will be targeted for their work. Mr. Laghmanai came to the U.S. with an S.I.V in 2013 and is now a U.S. citizen living in Houston with his wife and daughters, where he works for Refugee Services of Texas. His father, a doctor, worked for the Afghan government in clinics alongside U.S. service members.

A few months ago, the family in Kabul received a letter from the Taliban referring to Mr. Laghmanai’s work for the Americans and threatening his father. The family stopped going out in public and stopped sending its three youngest children, 8, 6 and 5, to school, fearing they could be kidnapped for retribution. This weekend, as the Taliban reached the city’s edges, Mr. Laghmanai’s mother locked herself at home with the children while his father and teenage brothers went into hiding elsewhere in Kabul.

The Taliban seized power in Afghanistan, with fighters entering the presidential palace, and gunshots broke out at Kabul’s airport as thousands tried to flee the country. The collapse of the Afghan government creates an uncertain future for civilians and challenges for the U.S. Photo: Wakil Kohsar/AFP The Wall Street Journal Interactive Edition

Representatives of the State Department and U.S. Citizenship and Immigration Services didn’t respond to questions about the family on Friday.

“I helped the U.S. for reasons, but I didn’t mean that it would turn against me and my family,” Mr. Laghmanai said. “I’m proud to be part of that mission with the U.S. armed forces, but I’m regretful that after doing all those things, the government is not taking that into consideration.”

Bashir Ghazialam, an Afghan-American and immigration attorney in San Diego, said he has been fielding constant calls from clients since Friday night. He said he is currently handling more than 50 visa applications for Afghans, as well as a dozen Special Immigrant Visas.

Bashi Ghazialam, an Afghan-American and immigration attorney in San Diego, said none of the Afghans he is helping obtain visas were allowed on flights out of Kabul on Sunday.

Photo: Law Offices of Bashi Ghazialam

He blamed the U.S. Embassy for having left so many Afghans behind. He said the embassy has a backlog of applications and described the application process as disorganized and lacking the proper staffing to adjudicate cases. The backlog has led to families being separated sometimes, as one family member may be approved for a visa while other relatives are stuck in the process.

Some of his clients had lined up visa interviews at the U.S. Embassy just before the Taliban seized Kabul but many were rescheduled in the past week to October. On Sunday night, Mr. Ghazialam told some of his clients to bring the paperwork they have and go to the airport. None of them were allowed on flights, he said.

The State Department didn’t respond to a request for comment on Mr. Ghazialam’s characterization. “Now we’re stuck in a situation where I have to comfort all these families,” he said.

Over the weekend, Mr. Ghazialam raced to complete an evacuation application from the State Department for an 18-year-old woman who lives in Kabul and has been awaiting a U.S. visa for five years.

The woman’s father, mother and six siblings moved to the U.S. in 2016 and settled in Walnut Creek, Calif., after her father was approved for a Special Immigration Visa. Her father, Mohammad Daud Intezar, worked as a driver for the U.S. Agency for International Development in Afghanistan and had received death threats for doing his job.

Mr. Intezar’s daughter didn’t immigrate with the rest of the family in 2016 because she was caring for her sick grandmother. A few months later, she tried to join them but was told that she needed to start a new family visa petition. The petition was approved in 2018, according to Mr. Ghazialam, but she is still waiting for a required interview with the embassy.

Mr. Intezar said he spoke to his daughter two days ago but hasn’t heard from her since because she has no electricity or internet.

“She said didn’t sleep at all when the Taliban came,” he said.

Suneeta, an Afghan immigrant living in Albany, N.Y., has been trying since 2014 to reunite with her four children, now ages 7 to 17, who she says are on their own in the Kabul area. This week, those efforts took a more panicked turn.

The children told their mother they went to the airport Sunday with their passports and documents from last year granting them humanitarian parole—which allows temporary admittance into the U.S. for emergency reasons—but they were turned away because they hadn’t secured a necessary visa. Suneeta, who is using only her first name so that her children won’t be targeted, heard they made it back to their apartment safely but weren’t doing well Monday, said Sara Lowry, an attorney with the U.S. Committee for Refugees and Immigrants who has been working with her.

A volunteer carried an injured man as other people waited at the Kabul airport on Monday.

Photo: wakil kohsar/Agence France-Presse/Getty Images

Suneeta’s husband worked with the U.S. military as an interpreter and team leader with the security forces at Camp Eggers. He disappeared in 2013 and is presumed dead. His disappearance makes his children ineligible for a Special Immigrant Visa, which would have required him to be the primary applicant. Suneeta became separated from her children in 2014, when her brother-in-law took them without permission to her husband’s family, shortly before she received legal authority to move to the U.S.

“I pray and hope to have them as soon as possible with me, together,” Suneeta said Sunday, through an interpreter. “They’ve already suffered a lot.”

Freshta Taeb, an Afghan-American activist, said Sunday she has been hearing from families trying to flee that are being turned away from the Iranian border at gunpoint and can’t find other ways out. She has had families tell her members of the Taliban have knocked on their doors asking how old their daughters are, if they are married and if they are in school.

“People who have daughters 12 and above are hiding them in basements,” Ms. Taeb said.

Mohammad Fawad Salihi said his sisters, both unmarried teachers in their 20s, are among the young women in hiding, terrified they could be forced into marriages with fighters.

Days ago, Taliban members paid a visit to Mr. Salihi’s father at his house in an area north of Kabul, saying they had heard Mr. Salihi was working for the Americans. No, his father told them; his son previously worked at the airport and then moved to Tajikistan. In reality, Mr. Salihi, 33, settled in Austin, Texas, in 2017, after working with the U.S. State Department in Afghanistan since 2009.

After the encounter with the Taliban, Mr. Salihi’s family back home split up, fearing for their lives. His mother stayed in their home to protect their belongings, hoping the Taliban wouldn’t hurt an older woman. His father and sisters went to hide with family in Kabul, bringing personal items they feared the Taliban would destroy, such as home videos from family weddings. Along the drive, Taliban members searched their car and smashed their computer and television, Mr. Salihi said.

Mr. Salihi’s wife, Nargis Salihi, was despondent watching her children Aleida, 4, and Arwin, 3, play in the family’s Texas apartment Sunday. Her husband was watching videos of Taliban fighters in the presidential palace where her brother worked as recently as yesterday. “Countries are going forward and we are going backward,” she said of Afghanistan.

Mr. Salihi, who is a driver for Uber and previously worked at a downtown Austin hotel, said he had held off on applying for visas for his family until he was financially stable enough to help support them.

“I wish I could have taken them out to a safe place, but it’s too late,” he said Sunday. “If something happens to them, probably always I’ll be saying, ‘This is all because of me.’”

Write to Elizabeth Findell at Elizabeth.Findell@wsj.com and Sara Randazzo at sara.randazzo@wsj.com