As an update to this, I read elsewhere that you have to "unload" these crates using the player logistic system to get the supplies out of them.
However, when I enable player logistics actions and select "unload", nothing happens. The player logistics actions also do not work with other items; I tried a few setups with cars and crates.
I have not placed the Player Logistics (disable) module. If I place that module to disable player logistics, then upon spawning the crew of the logistics helicopter bails out, the ammo box is immediately dropped to the ground, and the helicopter crashes. Somehow the crew survives the fall and proceeds to run on foot to the destination, at which point I get a message telling me that the supplies are delivered and the crew despawns.
I am using no mods other than CBA_A3 and ALiVE, and this was all done in a fresh editor scenario.