To encounter this bug, try the following:

• Spawn a truck and begin doing logistics runs between a supply point and a base
• Initially, the truck might be able to load / unload as expected
• Within about 10 runs, you ought to encounter the bug

It is hard to speculate why this happens. It might be because one has used or is using a captured vehicle.
Several people have mentioned this bug and when trying to fix it re-adjusted the position of the truck relative to the pickup/dropoff point. That doesn't solve it usually.
It might be the result of some sort of lag in the server, where the system does not yet realize/appreciate that a supply delivery/pickup unit is currently in range.
Trying to workaround the problem by standing between the vehicle and the supply point and then opening the tab and drag/dropping doesn't usually solve it either.
Sometimes, when one looks at the back of the truck, expecting to be able to see "unload supplies" in white ink, it remains greyed out for a while. One can wait 30 seconds in the hope that the system catches up with the situation and that sometimes does happen.

It is very demotivating for supply drivers to be unable to deliver their load due to this bug.