The problem with reading statutes is that what the statutes words say is not necessarily what the courts think those words say; or instantiate, to be a bit clearer. We think of statutes as imprecatory and even as imperative: the statute says it, and it is so. But what the statute says is determined by the court, sometimes by the case before the court. So, could Trump use the National Emergencies Act to build a border wall?
Probably not. Elizabeth Goitein makes an interesting, but wholly non-legal, argument that Trump has dithered for so long he cannot now claim an "emergency" exists which Congress doesn't have time to act on and so Trump must. Sounds interesting, but at no point does she ever cite a case where the courts have defined what an "emergency" is within the scope of the National Emergencies Act. I think her thesis is fundamentally sound:
Trump no doubt thinks he looks more reasonable if he gives Congress plenty of time to act before declaring an emergency. He might also think that Congress’s repeated failure to provide funds shows the need for emergency action. The truth is the exact opposite. By giving Congress time to definitively establish its unwillingness to fund the border wall, Trump is both taking away any legitimate justification for emergency action and proving his intent to subvert the constitutional balance of powers.
But, to put the matter bluntly, that's a common sense argument, not a legal argument (and there's a reason legal arguments don't always track with common wisdom, but I can't belabor it now). The court needs a bit more than "it's been too long" to go on. I think the case is clear that Congress has chosen not to act on the President's demand, and he can't simply declare an emergency in order to get his way. But will the courts agree? No one can say; aside from the Youngstown case, I've yet to read an analysis that cites any relevant case law, especially cases directly regarding the National Emergencies Act (which post-dates Youngstown by a few decades).
But the National Emergencies Act is not itself a blanket dispensation of authority. As Goitein points out:
The only powers the president can access during a national emergency are those Congress has granted. However potent some of these powers might be, the source of the president’s authority in all cases remains a legislative delegation—one that is granted in advance because true emergencies require immediate action. A president using emergency powers to thwart Congress’s will, in a situation where Congress has had ample time to express it, is like a doctor relying on an advance directive to deny life-saving treatment to a patient who is conscious and clearly asking to be saved.
Again, lovely analysis, but not really a keen legal argument. Still, the point in the first sentence is sound: the President only has the emergency powers the Congress grants; the President cannot grant himself the power to suspend Art. I, or Amendment 5, for that matter. And the National Emergencies Act explicitly recognizes this, because there are laws the President cannot suspend under the terms of the Act:
(a) The provisions of this chapter shall not apply to the following provisions of law, the powers and authorities conferred thereby, and actions taken thereunder:
(1) Chapters 1 to 11 of title 40 and division C (except sections 3302, 3307(e), 3501(b), 3509, 3906, 4710, and 4711) of subtitle I of title 41;
(2) Section 3727(a)–(e)(1) of title 31;
(3) Section 6305 of title 41;
(4) Public Law 85–804 (Act of Aug. 28, 1958, 72 Stat. 972; 50 U.S.C. 1431 et seq.);
(5) Section 2304(a)(1)  of title 10; 
This is the pertinent part of the text of one of those provisions (50 USC 1431, to be exact):
The President may authorize any department or agency of the Government which exercises functions in connection with the national defense, acting in accordance with regulations prescribed by the President for the protection of the Government, to enter into contracts or into amendments or modifications of contracts heretofore or hereafter made and to make advance payments thereon, without regard to other provisions of law relating to the making, performance, amendment, or modification of contracts, whenever he deems that such action would facilitate the national defense. The authority conferred by this section shall not be utilized to obligate the United States in an amount in excess of $50,000 without approval by an official at or above the level of an Assistant Secretary or his Deputy, or an assistant head or his deputy, of such department or agency, or by a Contract Adjustment Board established therein. The authority conferred by this section may not be utilized to obligate the United States in any amount in excess of $25,000,000 unless the Committees on Armed Services of the Senate and the House of Representatives have been notified in writing of such proposed obligation and 60 days of continuous session of Congress have expired following the date on which such notice was transmitted to such Committees. For purposes of this section, the continuity of a session of Congress is broken only by an adjournment of the Congress sine die at the end of a Congress, and the days on which either House is not in session because of an adjournment of more than 3 days to a day certain, or because of an adjournment sine die other than at the end of a Congress, are excluded in the computation of such 60-day period.
Sorry for the legalese, but the upshot is: the President will have to declare the border wall to be a matter of national defense (is Mexico invading us? Still, I can see the court declining to define that phrase narrowly); and cannot authorize contracts in excess of $25 million without notice to Congress and then the contracts (the language says "obligate" the US, not sign a single contract for that amount) and wait 60 days after that. This is not a measure meant to be used only for emergencies, but neither can it be suspended on the declaration of an emergency. And here is where the legal reasoning comes in.
A viable legal argument could be based on this provision of law, and the above provision of the National Emergencies Act, indicating there is no emergency because the wall itself is not a military necessity, and the President cannot enter into a contract to build it on an emergency basis. If he has to wait 60 days (at least), he can't claim construction is needed immediately, because he can't start construction immediately (as a matter of law; he can't start construction immediately as a matter of fact, either, but the courts will probably be more comfortable with what the law does, or does not, allow). If construction can't start the day of the declaration of emergency, and if Art. I establishes that Congress decides how to spend the money, then even if Trump says it must be done instanter, it can't be, as a matter of law; and so there is no legal reason to recognize a declaration of emergency under the circumstances, because the declaration would be seeking the power to enter into contracts post-haste, and that's a power explicitly excluded from the National Emergencies Act. Which means the President can't use the Act to declare an emergency to get his wall built, or even started, or even planned (it has to be planned, first, though Trump talks as if his words will cause the wall to spring forth in a "Fiat Lux" manner), because he can't use the Act to bypass the authority Congress has already granted to enter into contracts on behalf of the federal government. His declaration is, under the law, a nullity.
Off the top of my head that's an interesting legal argument, anyway. Moving away from legalisms a moment, the situation of a government shutdown (or the debt ceiling, an issue added to the mix by the Senator from South Carolina) raises another question: if the government can't spend money (or borrow money) Congress hasn't authorized (hence a shutdown or a debt crisis, respectively), on what Constitutional theory can the President do so?