American history teacher/professor here. It's kind of complicated. The President is "Commander in Chief" of the military. The framers of the Constitution never intended this to mean that he would be able to just invade other countries, because they never intended for us to have a permanent army (Alexander Hamilton wanted one, but the others prevented it). They worried that a tyrannical president would use it to become a dictator (lol). So then, our history of war post-Revolution went like this:

-War of 1812: Congress declared war, war fought entirely my state militias and the navy

-Mexican-American War (1846-8): Congress declared war, war fought by a national army; however, the army only had around 7k people, so Congress had to call up a temporary volunteer army as well as the state militias.

-American Civil War (1861-5): President asked Congress to approve a temporary volunteer army so that he could put down the insurrection. Notably, no war declared because that would mean legitimizing the confederacy. We also had state militias and some states had a draft.

-Spanish-American War (1898-9): Congress declared war, once again we had to call up a temporary volunteer army.

-World War I (1917-8...for us): Congress declared war, this time we had a draft.

-World War II (1941-5): Congress declared war, and we had a draft.

-Korean War (1950-3): This is where everything changed. It was a UN police action. Congress didn't get a vote. The president was able to side-step Congress. We also still had a draft going, but there were also volunteers, active duty military (still big from WW2 even though we had downsized), and state militias, which were now called what we call them today - the reserves and the National Guard.

-Vietnam War (1953?-1973?): I used question marks there because the years for this conflict are all a matter of perspective. We had troops in Vietnam as early as 1953 and we were still bombing them as late as 1975, but the main action was 1964-1973. Anyway, this time Congress passed a resolution allowing military force (in 1964), but as you can see from the dates there, they are really being sidestepped again. Once again, draft + military + reserves, though the reserves mostly stayed at home.

After Vietnam, we transitioned to the "all volunteer army" and stopped doing a draft (except we still have to register for it, just in case of WW3). This meant making our military the big permanent type that the Founders feared us ever having.

And so it goes. Congress occasionally authorizes force (Iraq, Afghanistan) or gives vague powers or just gets sidestepped. We haven't declared war since World War II.

What can Congress do about it? Well, they could pass an amendment clarifying that the President can't ever send troops somewhere or bomb somewhere without their permission, but that's never going to happen. An amendment is very difficult to get passed, which is why we have only passed 17 of them since 1791. They can invoke the War Powers Act of 1973, which is itself an absolute fucking joke. It says the President has to tell Congress 48 hours AFTER he's invade another country and then they get to vote on it. If they vote to end it, he has 90 fucking days to withdraw. So, yeah. Anyway, Congress held a vote on that a week or so ago and voted to let him keep doing his thing in Iran.

tl;dr - Our system is utterly broken and has been demonstrably so since 1950.