Any bases nearby? Maybe some could be stationed there
We have national guard who are trained truck drivers. Could they be used to transport material?
El Paso has Ft. Bliss for the Army, San Diego has Navy/Marine Corps, but most of the border is too far from existing bases to commute.
Truck driving, and other logistic support (helicopter/plane driving, medical, etc.) is what a lot of the Military being sent to support Border Patrol is doing. Also they do some high tech surveillance.
Some of the tactical, deployable, Combat Engineer units will be supporting as well, to install concertina wire. They can do some general construction work, but are not specialized for wall/fence building, so they would be less efficient, and produce more variable quality. Mostly their mission is to emplace or clear obstacles on battlefields, like minefields or roads.
They could do some, but it is really outside their wheelhouse. Civilian contractors will comply with all the local construction codes, and will produce reliable quality, or pay to fix it themselves.
The Army itself (Corps of Engineers) is managing the contracting for this construction. Just like when they build things on base, they have civilian professionals do the work, rather than soldiers.