Ammo boxes "reload" after five minutes. The first rank rifleman class has this unlocked,, so just drop a box every time it "reloads."
Although there is a slightly faster way to get this done, which is by unlocking the second rank for the support class (level three). The second rank for support class is able to drop both explosive ammo and weapon ammo crates, as well as a supplies box (the medics bandage box counts towards the achievement as well). This makes the achievement approximately twice as fast, while also contributing to that 5000 total supplies dropped.
Try to run within the radius of a manpower node too, as that will cut the "reload" time of all ammo boxes and supplies by half.