one of the worst affordable housing shortages in the island’s history.

The church where they are staying is home to the sole homeless shelter on the island. St. Andrew’s sits in a quiet corner of Edgartown, off the main drag where summer visitors feast on dripping ice cream and oysters.

The parish’s two buildings straddle a narrow, one-way street that on Thursday was crammed with police cruisers, immigration lawyers, translators, news cameras and a few dozen volunteers. Island residents stopped by throughout the day to donate food, clothes and shoes.

Just behind the parish, a crew of Brazilian laborers, a large immigrant work force on the island, was renovating a two-story home.

“We are meeting their needs for food, shelter, and we are definitely supplying them with a lot of love,” said Lisa Belcastro, the manager of the only homeless shelter on the island. “They need to be off island. Their immigration appointments are not here.”

Ms. Belcastro said they would be moved elsewhere, but when remains a question.

A man named Luis who had journeyed from the northern Venezuelan state of Lara with nine family members — including his wife, brothers and cousins — said they traveled through seven countries before making it to San Antonio and boarding the flight to Massachusetts.

Luis, who did not want to share his surname out of concern for his safety, said that after spending three days in San Antonio, his family also encountered a blonde woman in the street who said her name was Perla. The woman, he said, told them she was affiliated with a foundation that helped transport refugees to sanctuary cities and could book them on a flight to Massachusetts. One caveat: They could not take any pictures.

When the pilot announced they were about to land on Martha’s Vineyard, the migrants were shocked, Luis said: “That’s the first time we heard that.” He broke down in tears when he got to the island.

Like other migrants, his family has paperwork from border authorities informing them to report to an immigration office within 15 days, and the man said he feared that his family had been tricked into flying to the island to then be deported.

Venezuelan migrants surrender to border officials once they cross into the United States, and later request asylum. Since the United States has no diplomatic relations with their home country, they are generally allowed to stay, rather than face expedited deportation like many migrants from Mexico and Central America.

After reaching their destinations, the migrants can remain in the United States for months or even years while they await the outcome of their asylum cases, and they are allowed to work while they pursue asylum claims. But they must comply with instructions to appear in court or check in with immigration authorities, or they jeopardize their cases.

Back in San Antonio, Luis Amador Castillo, 34, recalled that he and a friend, Francisco, had crossed the street to the McDonald’s near the shelter.

A woman handed them bottles of water and gift cards from the restaurant. Are you interested in a free hotel stay and a free flight to Massachusetts? she asked with a smile.

There are people waiting for you with food, shelter and ready to help you with whatever you need, the woman told them, Mr. Castillo recalled. Francisco jumped at the idea.

Mr. Castillo said he politely declined because he intended to stay in San Antonio for the time being and later travel to Houston to find work, he said.

He recalled that the name of the place she mentioned sounded funny at first. “She said it was Massa-something,” Mr. Castillo recalled on Thursday while standing outside a shelter run by the city.

Steps away from him, Xavier Lopez, 20, and his girlfriend, Maria Perez, 21, said they regretted missing a free flight. The pair left Venezuela in late May with their 2-year-old son, Santiago.

“We don’t have money to go anywhere. We are going to stay here until we figure out how to find transportation,” Mr. Lopez said. “If I knew there was a person offering flee flights to Miami, that would be ideal for us.”

Miriam Jordan reported from Los Angeles, and Remy Tumin from Edgartown, Mass. Will Sennott contributed reporting from Edgartown, and Edgar Sandoval from San Antonio.

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