No, they do not. "The high court has said Congress is not a law enforcement agency, and cannot investigate someone purely to expose wrongdoing or damaging information about them for political gain. A subpoena must potentially further some “legitimate legislative purpose,” the court has said."
there has to be a legislative purpose. They are NOT the Justice Department.
The immunity of the President’s immediate advisers from compelled congressional testimony on matters related to their official responsibilities has long been recognized and arises from the fundamental workings of the separation of powers.
This immunity applies to former senior advisers such as the former White House Counsel.
Accordingly, the former Counsel is not legally required to appear and testify about matters related to his official duties as Counsel to the President.
The President does not waive an adviser’s immunity from compelled congressional testimony by authorizing disclosure of any particular information.
The disclosure’s impact on executive privilege does not ultimately bear on any underlying immunity from compelled testimony.
Because Congress may not constitutionally compel the former Counsel to testify about his official duties, he may not be civilly or criminally penalized for following a presidential directive not to appear.
The same rationale applies equally to an exercise of inherent contempt powers against a senior aide who has complied with a presidential direction that he not provide testimony to a congressional committee.
On April 22, 2019, the Committee on the Judiciary of the House of Representatives subpoenaed Donald F. McGahn II, the former Counsel to the President, to testify about matters described in the report of Special Counsel Robert S. Mueller, III.
You have asked whether Mr. McGahn is legally required to appear.
We provide the same answer that the Department of Justice has repeatedly provided for nearly five decades: Congress may not constitutionally compel the President’s senior advisers to testify about their official duties.
This testimonial immunity is rooted in the constitutional separation of powers and derives from the President’s independence from Congress.
As Attorney General Janet Reno explained, “subjecting a senior presidential advisor to the congressional subpoena power would be akin to requiring the President himself to appear before Congress on matters relating to the performance of his constitutionally assigned executive functions.” Assertion of Executive Privilege with Respect to Clemency Decision, 23 Op. O.L.C. 1, 5 (1999) (“Reno Opinion”).
Congress may no more summon the President to a congressional committee room than the President may command Members of Congress to appear at the White House.
https://www.justice.gov/olc/file/1164186/download