I see this as a very difficult question. I start from the premise that the declare war clause gives Congress exclusive power to decide whether the United States will initiate war (although that power is delegable to the President). Applying this rule to the situation in Iraq is, to say the least, tricky. At least 5 questions must be answered:
(1) Can the U.S. be at war with ISIS? That is, is ISIS enough of a state-like entity that engaging it militarily is a "war" in the constitutional sense? I would think that helping an ally suppress rebel groups within its borders is often not a war, because the rebel groups are too unstable and disorganized to look anything like an entity that could be one side of a war. On the other hand, I assume that an entity need not be a "state" in the formal modern international law definition in order to be part of a war. (Attacking Hamas in support of Israel, for example, would seem to initiate a war with Hamas, even if Hamas is technically not a state).
(2) If the U.S. can be at war with ISIS, is the U.S. at war with ISIS? Perhaps infrequent and defensive airstrikes, designed to protect U.S. personnel in Erbil and/or to assist evacuation of civilians but not to fundamentally threaten ISIS's control of its territory or its military power does not count as warfare. (In contrast, the 2011 Libya campaign, designed at minimum to prevent the Libyan government from using military power to destroy opposing forces, and apparently more broadly to assist in driving it from power, should count). To be sure, it's not clear that the U.S. operation against ISIS is so limited.
(3) If the U.S. is at war with ISIS, is this a new war? Perhaps the ISIS conflict is just the latest iteration of the conflict with Sunni extremists that the U.S. fought in 2007-08. If so, it's an extension of the prior, congressional authorized conflict in Iraq, and therefore does not need any further authorization. (See also # 5(a) below, although these are not exactly the same questions).
(4) If this is a new war, did the U.S. start it? Here the key question is not who started the shooting, but who first showed a determination to settle differences between ISIS and the U.S. by force? Perhaps ISIS initiated a state of war with the U.S. by making bellicose statements such as (apparently) saying it would plant its flag on the White House. In my view (and Alexander Hamilton's), when another country begins a war with the United States, the President does not need Congress' approval to respond.
(5) If the U.S. started a war with ISIS, did the President acting without Congress' approval? (a) Perhaps authority comes from the authorization to use force in Iraq, which is not limited to any particular adversary or time period (and which apparently allowed the U.S. to fight a previous round of battles against Sunni rebels in western Iraq) — Jack GOldsmith makes the argument here. (b) Perhaps authority comes from the authorization to use force against al Qaeda and its supporters, since ISIS (maybe) started as an al Qaeda off-shoot. (Also (c), perhaps authority comes from the war powers resolution, although I discount that argument).
Each of these questions must be answered "yes" to make the President's action unconstitutional. I am not confident about the answers to any of them.
Posted at 6:40 AM