Is there a way to get the ammo from a gun without picking it up?
I noticed this as well. Sadly, it seems like there is no way to get weapon ammo any other way (besides picking up the dropped ammo boxes of enemies of course).
The only ways to get ammo at all is by running over the dropped ammo boxes, or buying ammo from some shops (which is extremely expensive), or picking the gun up. I guess they did it so that you have at least a little bit of ammo when you pick up a gun, without making it to plentiful, I guess they just didn't realise people would do that. Your choice seems to be convenience+time vs. extra ammo.