There is a certain amount of deliberate ambiguity to Article I Section 8 -- take a look at e.g. https://avalon.law.yale.edu/18th_century/debates_817.asp and you can see that the wording "The Congress shall have power to [...] declare war" was revised from "make war" in the earlier drafts. Madison clearly felt that Congressional authorization was on some level required to conduct a war, but that the Executive should be free to act quickly in self-defense, e.g. to repel an invasion.
Like a lot of things, this seems to fall under the idea of "convention" and not law. This has been an ongoing problem in recent times. Practically speaking, Congress is a rubber stamp for matters of war.
Theoretically, they can withhold funding from the military. But seeing as the Treasury Department falls under the President, it's unlikely they can actually do that.
"Withhold funding from the military" it's actually better (or worse) than that. The US doesn't have a standing army technically speaking, it cannot constitutionally. The post war military has been continuously reauthorized twice yearly since the end of world war 2. Congress can refuse to reauthorize continuation of it and the treasury can't do anything about it.
Legally though the power of the purse also lies with Congress. If we're throwing in extra constitutional actions all bets are off though because we're no longer bound by the rules of what is allowed and it's down to the old power of might making right.
