This is an interesting question that raises many of the same issues as the US involvement in Libya and the proposed involvement in Syria. The possible bombing of ISIS, of course, raises additional issues.
There are two possible bases for action by the executive without congressional authorization. First, the executive might rely on statutory authority that Congress provided in the past. There is the 2002 Iraq AUMF (Authorization for Use of Military Force), which provided that “The President is authorized to use the Armed Forces of the United States as he determines to be necessary and appropriate in order to . . . defend the national security of the United States against the continuing threat posed by Iraq.” Jack Goldsmith argues that this provision authorizes the proposed US bombing, based on a textual analysis. Wells Bennett contends that this textual language should be limited by the context of the AUMF – focused as it was on Saddam Hussein – as reflected in its preamble and other provisions. Interestingly, Robert Chesney argues that the 2001 AUMF linked to al Qaeda might provide the authority, despite the disagreements between ISIS and al Qaeda.
The second possible bases for unilateral executive action is Article II of the Constitution – specifically, the Commander in Chief and Executive Power clauses. Mike Ramsey powerfully argues on originalist grounds that the President would not have the authority to take action if ISIS if it were considered a state. But since ISIS is a non-state actor, he only tentatively leans toward saying that the President lacks the authority.
Another argument that denies that the President has the power to exercise unilateral execution action is Jack Goldsmiths’, but Mike Ramsey raises serious questions about it.