Article IV Section 4 of the US Constitution states that the US government must "protect" each State from "invasion". It goes on to include protection from "domestic violence", but that requires action from the Legislature, unless the Legislature can't convene, then the Executive branch must take action.
It does not say, "The US government may protect the people if it wants to. And, if the Legislature refuses to, then the President must allow anyone and everyone to just destroy whatever they want to." It also does not say, "...unless they really, really want to come into the country, and that's why they're invading." But, that's how House Democrats would like to have it interpreted.
What's happening at the southern border is an "invasion"—people from the outside coming in by force. The Constitution does not specify that the invasion must be a sanctioned, deputized, funded military force operating at the behest of a recognized State. Any and every kind of invasion must be stopped, by Legislature or otherwise.
The Legislature is only required for situations of domestic violence. But, even then, if the Legislature can "convene"—and it can—but disobeys this Constitutional requirement, that could be cause for an action of impeachment because they would be in violation of their oaths of office, to support the Constitution. Then, the power to stop domestic violence would fall to the Executive branch, namely the president.
Trump is well within his powers to declare an emergency and take executive action, but he might be Constitutionally required to begin impeachment proceedings against Congress if the border situation is regarded as domestic, not an "invasion" from non-US citizens. So, claiming that Congress is needed to build the wall would actually be an argument to indite Congress.
As for citizenship by birth, that applies only to children of parents "subject to the jurisdiction thereof". Amendment XIV does not say, "Anyone can run from the police, sneak into the country, have a baby, then demand citizenship for that baby." But, that's how Congressional Democrats would like to have it interpreted.
The US is about to rediscover its Constitution, the document that united our nation at its founding. That could redefine the entire playing field of elections in the future.