It is not the first time that questions have been raised about how Ms. Trump and Mr. Kushner have used private communications in their government roles. Both have used personal email accounts to communicate with other White House officials and cabinet secretaries about official business. But the irony of the new accusations was savored by Mrs. Clinton’s former aides.

“For his endless faults, Jared should at least know what chutzpah means,” said Philippe Reines, a top adviser to Mrs. Clinton, referring to the Yiddish word meaning “gall.”

“And while there’s no specific Yiddish word for continuing the practice even after the Democrats took control of the House, ‘mashugana’ fits,” Mr. Reines said, referring to the word meaning “crazy.”

Mr. Cummings based his findings about Ms. Trump and Mr. Kushner on what the committee had been told by the couple’s lawyer, Abbe Lowell. But Mr. Lowell insisted in a letter to the committee that Mr. Cummings was twisting his statements and that he had not said what had been claimed.