No, they don't.
Trump can't just take what isn't his, you ******* moron.
Presidential Records Act (44 U.S.C. Chapter 22)
(g)(1) Upon the conclusion of a President’s term of office, or if a President serves consecutive terms upon the conclusion of the last term, the Archivist of the United States shall assume responsibility for the custody, control, and preservation of, and access to, the Presidential records of that President. The Archivist shall have an affirmative duty to make such records available to the public as rapidly and completely as possible consistent with the provisions of this chapter.
(2) The Archivist shall deposit all such Presidential records in a Presidential archival depository or another archival facility operated by the United States. The Archivist is authorized to designate, after consultation with the former President, a director at each depository or facility, who shall be responsible for the care and preservation of such records.
(1) The term "documentary material" means all books, correspondence, memoranda, documents, papers, pamphlets, works of art, models, pictures, photographs, plats, maps, films, and motion pictures, including, but not limited to, audio and visual records, or other electronic or mechanical recordations, whether in analog, digital, or any other form.
(2) The term "Presidential records" means documentary materials, or any reasonably segregable portion thereof, created or received by the President, the President’s immediate staff, or a unit or individual of the Executive Office of the President whose function is to advise or assist the President, in the course of conducting activities which relate to or have an effect upon the carrying out of the constitutional, statutory, or other official or ceremonial duties of the President. Such term--
(A) includes any documentary materials relating to the political activities of the President or members of the President’s staff, but only if such activities relate to or have a direct effect upon the carrying out of constitutional, statutory, or other official or ceremonial duties of the President.