[Congress shall] provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions; Art I Sec. 8.
Section. 2. The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States;
This last clause is key. The President has authority to call up the militia, and call-ups of militia are for emergencies, not to do the job of the regular military, which is provided for elsewhere. So to call up the militia is to declare an emergency.
So can the President declare an emergency without calling up the militia? All U.S. citizens, including government employees and contractors, are militia. Directing them to reallocate funds for defense is to act within that power. No special statutory authority is needed.
So are entries into the U.S. without consent an invasion? Yes. Any such trespass is an offense against the law of nations, which Congress has the power to define and punish. They have done that, although first-time simple entry is merely a "deportable offense", a kind of misdemeanor. However, reentry after having been deported is a felony.
It does not need to be an armed force to be an invasion. A child chasing a butterfly across the border is an invader. It also doesn't matter whether the invaders are, or can be expected to be, criminals. Peaceful people seeking work are also invaders, if they enter without consent.
So is the situation on the southern border an emergency? If it were only a few a day, no. But thousands flooding the border, faster than they can be managed, is an emergency.
Does it matter that the thousands are seeking asylum? No. U.S. law only recognizes political asylum, not economic asylum. Most of those thousands are economic refugees. If they are fleeing criminals or corrupt officials, then they have the duty to fight in their own countries, not in ours.
What is the President's alternative? He could station troops along the border with orders to repel invaders with deadly force. He could erect gun turrets every few hundred yards. That would be more expensive than a wall. Do opponents of a wall really want invaders to be repelled by automatic weapons? Democrats would not get many votes from those.
A clue to the ambitions of many illegal entrants can be found in an exchange with a border rancher, who ordered them off, saying "This my land!" One of the invaders replied, "No, this is our land."\
In other words, the invaders are trying to conquer the U.S., a few acres at a time. They already have gained de facto control over large areas, extorting from Americans and calling it "rent". That is largely how the cartels make much or most of their money in their home countries. Selling narcotics is becoming less important.
So the real issue is, do we want our country to be ruled by the criminal cartels?