A man posing as a pest exterminator tried to gain access to a San Diego hotel that operates as a shelter for migrant families. The next day, a woman showed up claiming to be an immigrant in need of help. Workers at the shelter, run by Catholic Charities, turned away both impostors.

Three days later, menacing calls began pouring in to the staff. Voice mail left for the chief executive called him “scum” and “not really Christian.” A woman left another staff member an expletive-laced message about Catholics. She claimed that the nonprofit was flying migrants all over the country and profiting from an illegal operation.

The bogus exterminator was James O’Keefe, the right wing-provocateur who used to head Project Veritas, a group known for trying to entrap political opponents by using disguises and concealed cameras. The deluge of vitriol ensued after Mr. O’Keefe began posting videos on X in March claiming that the shelter was an illegal holding site for women and children and speculating, without evidence, that they had been trafficked.

For decades, Catholic Charities and other faith-based organizations have played a crucial role helping federal authorities and local governments manage influxes of migrants. Their work has been funded with bipartisan support in Congress, even through the presidency of Donald J. Trump, who often vilified immigrants.

But after President Biden took office in 2021 promising a more humane approach to migration, these faith-based groups have increasingly become the subjects of conspiracy theories and targets for far-right activists and Republican members of Congress, who accuse them of promoting an invasion to displace white Americans and engaging in child trafficking and migrant smuggling. The organizations say those claims are baseless.