If the Military spends money on the wall, they would first have to pick something in their budget to take the money away from.
Congress could simply deduct the amount spent from their next budget, leaving them unable to pay for whatever they took the wall money from.
So although the Military could execute the mission, they really would need the money from Congress, or it could hurt them badly elsewhere.
One significant exception that really is in play now, is that portion of the border under direct Military control - the Barry M. Goldwater Range in SW Arizona, which includes over 30 miles of International border.
Secretary of Defense Mattis is considering whether or not to pay for new Trump-style border barrier along that stretch - almost half a billion dollars.
My guess is that the decision will be delayed, until they see how the mid-term elections turn out.
The Tohono O’odham Reservation (Managed by the Department of the Interior) is not far from there, and accounts for a dis-proportionally high percentage of drug seizures. The Department of the Interior controls Indian Reservations, National Parks, and Bureau of Land Management lands along a lot of the border. Although not discussed in this article, the Department of the Interior could also build border barrier, if funded.
There is no more delay. The president said “September 30”— THIS yEAR, or he would shut down the government, It is crunch time. If he caves again, this party is in real trouble in November. We are not interested in a 30 mile strip on a military reservation. We want all 2000 miles with a massive and effective border WALL manned with thousand of agents using drones and tower and atvs. 63,000,000 clearly voted for the wall. We want the WALL!!