The core order and authorization as stated in the order is:
by virtue of the authority vested in me as President of the United States, and Commander in Chief of the Army and Navy, I hereby
authorize and direct the Secretary of War, and the Military Commanders
whom he may from time to time designate, whenever he or any designated
Commander deems such action necessary or desirable, to prescribe
military areas in such places and of such extent as he or the
appropriate Military Commander may determine, from which any or all
persons may be excluded, and with respect to which, the right of any
person to enter, remain in, or leave shall be subject to whatever
restrictions the Secretary of War or the appropriate Military
Commander may impose in his discretion.
A related order, 8972, uses the same phrase for a related purpose:
by virtue of the authority vested in me as President of the United States, and Commander-in-Chief of the Army and Navy of the United
States, I hereby authorize and direct the Secretary of War
Truman's Executive Order 10340 similarly ordered
by virtue of the authority vested in me by the Constitution and laws
of the United States, and as President of the United States and
Commander in Chief of the armed forces of the United States...
to seize steel plants, in light of an strike. In that particular case, SCOTUS ruleds in Youngstown Sheet & Tube Co. v. Sawyer, 343 U.S. 579 that he did not actually have that authority. There is a difference between claiming an authority and having that authority validated by the courts. In Hirabayashi v. United States, 320 U.S. 81, one of the holdings was that
By the Act of March 21, 1942, Congress ratified and confirmed
Executive Order No. 9066, and thereby authorized and implemented such
curfew orders as the military commander should promulgate pursuant to
that Executive Order.
It was within the constitutional authority of Congress and the
Executive, acting together, to prescribe this curfew order as an
emergency war measure
(Note that the order preceded the congressional authorization referred to: the curfew order is not part of EO 9066, but is a consequence of it).
In general, you cannot know if an order is really authorized by Congress or the Constitution until the Supreme Court rules on the question. This has happened only a handful of times, notably 5 times in Roosevelt's case.