At one point, the Obama Administration wanted a Congressional authorization for their war against ISIS. They openly bragged about the vague language allowing them to do more or less anything, and never got Congress to agree. Still, the war continued.
Instead of worrying about what the legal justification of the war might well be if the law was different, however, people are increasingly asking what the current justification is, particularly as the Syrian part of the war broadens, with more targets and more enemies. The answer is complicated, and unsatisfying.
When they’re attacking al-Qaeda’s Nusra Front, the justification is the 2001 AUMF for the Afghan War, because it’s al-Qaeda. When they’re fighting ISIS, it’s sort of the same, on the grounds that ISIS is sort of al-Qaeda, even though they’re actually not.
Then there’s talk of attacking Syria’s own military. The justification there is even less obvious, and the administration is claiming Article II of the US Constitution allows them to defend “assets,” and that they consider the rebels they’re trying to create in Syria assets, and therefore they can do whatever they want to any conceivable enemy of those rebels.
Constitutional scholars are spurning this argument, noting that Article II isn’t nearly specific enough to justify anything, and that with such an overbroad interpretation the US could declare whatever it wants in whatever country to be an “asset” and then fight a war on that basis.