Tens of thousands of families around the world are at risk of losing a rare opportunity to immigrate to the United States.

For the past three decades, the Diversity Visa Program has awarded a path to legal permanent residence to about 55,000 people each year from countries with low levels of immigration to the U.S. Each applicant has a less than 1% chance of winning a green card.

The U.S. government must process the applications of lottery winners before the end of the fiscal year on Sept. 30, or else the winners will lose their shot at a green card. As of June 2021, because of what the Biden administration says are Covid-19-related restraints, the State Department had only processed about 3% of the total 55,000 visa applications. Several ongoing federal lawsuits ask the State Department to reserve diversity visas to be processed after the coming deadline.

For now, Fatima Gibreel’s hope has an expiration date. She is a 29-year-old Sudanese farmworker who learned she won the visa lottery in May 2020.

“I saw I won…that was the first time my sisters and I had any hope. We didn’t have any for a very long time,” said Ms. Gibreel.

A State Department official said 2021 winners are welcome to apply again next year.

But along with the slim odds for any applicant to be chosen, the lottery isn’t free. There is a mandatory $330 nonrefundable fee to be interviewed at the U.S. Embassy, plus medical examination and translation costs. Ms. Gibreel says that even if she miraculously won the lottery a second time, the cost would disqualify her family from receiving a green card.

“We sold the house that our father left us before he died,” Ms. Gibreel said. “The house was sold in the hope that I would go to America and improve my family’s situation.”

Ms. Gibreel said that she and her sisters earn less than a dollar a day, spraying pesticides on local cotton and wheat farms.

Last year, the pandemic closed consulates across the world, bringing immigration to a grinding halt. Then-President Donald Trump, who once told a group of lawmakers that the diversity visa lottery brings over people from “shithole countries,” froze diversity visas and some other visa types in the early months of the pandemic, citing public health concerns surrounding the spread of Covid-19. A federal judge partially reversed the diversity visa order in September 2020.

After pushback from federal courts, the Trump administration compromised: The State Department released a new set of visa processing priorities, placing diversity visas in the bottom tier. And despite President Biden’s campaign promises to protect and even expand the diversity lottery program, those visas remain at the bottom. According to the federal government’s own court filings, in March and April of this year, the State Department adjudicated 61% of its normal capacity across all visa categories, but for the same period, only 8% of its normal capacity for diversity visa applications.

The diversity lottery program has relatively few friends in Congress. Republicans say it brings mainly low-skilled immigrants and poses potential security risks. Democrats agreed to kill the program in their 2013 overhaul of the immigration system that passed the Senate, but it was never enacted into law.

GOP criticism of the program increased in 2017 when it was revealed that Sayfullo Saipov, an Uzbek immigrant who drove his truck into a bike path in New York, killing eight people, came to the U.S. through the diversity visa lottery program. Mr. Trump called for the program’s repeal.

But the Congressional Black Caucus has championed the program, calling it the primary source of African immigration to the U.S. Although the lottery was originally created to benefit Irish immigrants, the program has more recently benefited immigrants mostly from Africa and some countries in Eastern Europe and Asia. In the most recent lottery, the results of which were announced in July, the countries with the most green-card winners included Egypt, Sudan, Russia, Algeria and Uzbekistan, according to State Department figures.

Curtis Morrison, one of the immigration attorneys leading the current litigation against the State Department, says his clients have sold homes, delayed college and taken out loans, all in the hope of coming to the U.S.

“What [the U.S.] did is we went and told this group of people, you are selected, you won,” said Mr. Morrison. “And then the Biden administration, previously Trump, decided ‘no, we’re just not going to go through with it because of Covid.’”

A State Department official said the Biden administration is hampered by ongoing public health concerns.

“The Department is making every effort to process as many Diversity Visa cases as possible, consistent with other priorities, despite the severe operational constraints and backlog resulting from the Covid-19 pandemic,” the official said in an emailed statement.

Immigration attorneys like Mr. Morrison suspect that the pandemic is only one barrier in the diversity visa process. The lottery program is administered by the Kentucky Consular Center, a small division of the State Department based in Williamsburg, Ky. A State Department official said the KCC call center has been closed since September 2020, but staff is still communicating with winners via email.

Brandon Goh is a 2021 diversity visa lottery winner from Malaysia.

Photo: Brandon Goh

“The Kentucky Consular Center (KCC) is experiencing an unprecedented volume of inquiries and is striving to respond as promptly as possible,” the official said.

Callers to the KCC reach a voice message that redirects winners to an email inbox, which State Department attorneys admit was full and not accepting new messages at one point last May.

“We only informed [the KCC] about the inbox because we could reach out to their lawyers,” Mr. Morrison said. “There’s literally no other way to contact them. What’s an applicant without internet access supposed to do?”

Brandon Goh is a 2021 lottery winner from Malaysia. The U.S. Embassy in Kuala Lumpur has been responding to emails, Mr. Goh said.

“But even though my consulate has capacity, it’s being wasted because the KCC doesn’t send cases to the consulate,” Mr. Goh said.

The embassy wrote to Mr. Goh saying it can’t help process his visa because that is the purview of the KCC. But Mr. Goh has struggled to reach anyone at the center or schedule an appointment.

On Thursday, KCC Director Morgan Miles released a declaration stating that the State Department could schedule additional lottery visa interviews on an ad hoc basis, if consulates inform the KCC that they have the capacity to conduct interviews.

Charles Kuck, the attorney helping represent Mr. Goh, sees this latest statement from the KCC as a potentially promising sign. Mr. Goh is crossing his fingers, hoping the embassy in Kuala Lumpur will press the KCC to schedule his interview in the coming weeks. He has contacted the consulate about Mr. Miles’ latest statement and hasn’t heard back.

Back with her sisters in Sudan, Ms. Gibreel says she is exhausted from living in limbo. She has also repeatedly emailed the Kentucky center, and never heard back.

“I just want my interview, that is all,” she said. “I don’t have hope anymore.”

—Michelle Hackman contributed to this article.