Two days after Mr. Kinzinger called for removing Mr. Trump from office following the Jan. 6 riot at the Capitol, 11 members of his family sent him a handwritten two-page letter, saying he was in cahoots with “the devil’s army” for making a public break with the president.

“Oh my, what a disappointment you are to us and to God!” they wrote. “You have embarrassed the Kinzinger family name!”

The author of the letter was Karen Otto, Mr. Kinzinger’s cousin, who paid $7 to send it by certified mail to Mr. Kinzinger’s father — to make sure the congressman would see it, which he did. She also sent copies to Republicans across Illinois, including other members of the state’s congressional delegation.

“I wanted Adam to be shunned,” she said in an interview.