While I don't think it would prevent our troops from having foreign-produced trucks in theater, we can't affordably procure such trucks thanks to the Chicken Tax. I would also guess that giving a DoD contract to Toyota for a truck that may not be registrable in the US would also face institutional resistance. https://en.wikipedia.org/wiki/Chicken_tax
The military has an incentive to ensure there are plenty of Americans who know how to design and manufacture things. A truck and a tank have a lot in common - if war breaks out we want the ability to take people of of trucks and get them making things the military needs. This is the same reason the Navy has for building ships in the US even though they can be done other places cheaper.
You'd think the biggest war machine on the planet would benefit from economies of scale by now. If they want to stay sharp they could build commercial ships between the ocassional war ship.