The South Lawn of the White House was decked out in New York City paraphernalia for President Trump’s first congressional picnic: food carts, boat racing and a Central Park-like carousel, all in honor of the president’s hometown.

Chris Collins, a Republican House member from western New York, should have been in his element. He had shot to fame the year before as the first member of Congress to endorse Mr. Trump for president. He had since channeled that fame into a spot on Mr. Trump’s transition team and frequent appearances on cable news.

But on that early summer evening in 2017, Mr. Collins seemed distracted. Standing apart from the festivities, he paced back and forth on the White House lawn, his cellphone pressed to his ear. When Jared Kushner, the president’s son-in-law, strode by, Mr. Collins did not seem to notice.

On the other end of the call was Mr. Collins’s son. And the six-minute exchange that transpired would form the basis of an insider trading investigation that led to Mr. Collins’s indictment, and to his resignation and guilty plea last week. His son has also pleaded guilty.