The comics have fudged the issue for decades. Marvel now describes it as an “extra-government” body, although many takes on the organization have clearly emphasized its international character. Yet U.S. presidents have fired S.H.I.E.L.D. directors (Fury, Tony Stark/Iron Man) and appointed others (Norman Osborn/The Green Goblin, incumbent Steve Rogers/Captain America) — although that might operate by an informal international understanding, much like the U.S. appoints the director of the World Bank.
It’s hard to imagine a truly international organization with international stakeholders permitting a guy named “Captain America” to run their elite strike force, though.