Miguel Ángel García, a house painter and father of four from Mexico, was picked up a few weeks ago by U.S. Immigration and Customs Enforcement officials on a routine traffic stop, family members say, after living without documentation in the Dallas area for about 20 years.

Then on Wednesday, he was shot four times, including in the neck, while shackled in the back of a government transit van.

Mr. García has yet to be publicly named by the federal government as one of the two detainees in critical condition after a gunman opened fire at a Dallas ICE facility. Three people familiar with the investigation confirmed Friday afternoon that Mr. García was one of the injured, along with Jose Andres Bordones-Molina of Venezuela. Norlan Guzman-Fuentes of El Salvador was killed, they said.

Federal investigators said Thursday that they were not releasing the names of the victims pending notification of their families, but family members are beginning to speak up, to plead for financial help, to tell of their struggles to get information from the government and to share their shock at finding the victims of the shooting still shackled to their hospital beds.

“Seeing him handcuffed like that, it just didn’t feel right,” said Mr. García’s brother, who asked he not be identified because he is in immigration proceedings and has not yet obtained legal status. “No one ever imagines something like this could happen.”

Surveillance video, released by the Dallas Fox News affiliate, confirmed that detainees aboard the white transit van were shackled and handcuffed as they fled the van for their lives, shuffling and tumbling as federal agents pointed them to safety.

Mr. García is one of four brothers who arrived in Texas with their parents without legal status more than two decades ago, when the boys were between the ages of 10 and 15, his family said. They left behind their native Mexican state of San Luis Potosí. Mr. García started painting houses as soon as he was old enough to work.

His mother was deported to Mexico earlier this year, when federal immigration officers combed their neighborhood in search of other men. Mr. García landed in the federal immigration system in August after he was picked up for driving while intoxicated and evading arrest in Tarrant County, according to court records. He had a previous charge of providing false information.

His relatives have since started fund-raising in hopes of covering his medical expenses. A GoFundMe web page says he was the sole provider for his wife, who has a baby on the way.

“He was a victim of the ICE facility shooting in Dallas,” Gabriela Gauffeny, Mr. García’s sister-in-law, wrote on the fund-raising page, in English and Spanish. “He is in grave condition.”

The government’s delay in publicly releasing the names of the detainees hurt and killed in the shooting has prompted a vigil and calls for transparency from immigrant rights groups and Latino community leaders. Officials at ICE and the Department of Homeland Security did not return requests for comment on Friday.

Mr. García’s family said they struggled to obtain information about his whereabouts, and that a few of his relatives were initially turned away from visiting him in the hospital.

On Friday, the daughter of another detainee trapped inside the van said her father miraculously survived the attack. He had appeared for a routine appointment at the ICE check-in office when he was apprehended and taken to the back of a vehicle with other detainees.

He later called his wife from another ICE facility and said that he was in shock the moment bullets came flying, according to a recollection from his daughter, who also asked not be identified for fear of retaliation from federal immigration authorities.

“He felt a bullet touch his shirt, but thankfully was not hit,” his daughter said in Spanish. “He called my mother to tell her he was still detained.”

Like Mr. García’s family members, she said she has not heard from ICE officials or been offered any assistance, but, she added, there was no need to complain, “because we know we are not going to be heard.”

Hamed Aleaziz contributed reporting from Washington. Susan Beachy contributed research.