Last I checked, Section 3, Clause 5 of the US Constitution required the President to "take care that the laws be faithfully executed." Since illegal entry into the country is, well, illegal ... and it's a Federal-jurisdiction issue, as the illegal-entry is into the country, not a State ... seems to fall well within the purview of the Executive Branch. The means of execution are largely at the discretion of the President.
But does it say that to faithfully execute that law, a wall must be built?