I had one soldier fill up, and there a still a couple items left in a box. I stand next to it with another soldier and can’t figure out how to access it. What am I missing/
Just press the inventory button, or ‘i’
You need to come by and use inventory manually.
Its no longer glowing and there is no plus will reward, but you can pick whatever is remaining.
If you have enough action points. Quite necessary on scavanging.
To expand it a bit, on higher difficulties nice equipment can be found in boxes,
that leads to reverse engineering.
It’s worth noting this is exactly what you use to pick up items off the ground as well, if you were not already aware (based on your original question).