This solution requires you to make good (possibly excessive) use of the Soldier class's Ammo Sharing ability. Each time you hand out an ammo buff, it takes one of your Supply pips (shown on the bottom left of the HUD)
The achievement asks you to spend your last Supply pip on handing out an Ammo Buff. Of course, to do this, you must be a Soldier.
Another thing to note - the achievement states "..refill an teammates ammo RATHER THAN YOUR OWN" . So if you're running around dispensing ammo like a warfaring PEZ dispenser, it's worth remembering that you should empty a clip or so, just to make sure you require an ammo buff yourself.
This achievement is most easily attained with a character who does not possess the following abilities:
Universal:
Supply Max Increase
Resupply Rate Increase
These perks deal with increasing the rate and capacity of your supply pips. Obviously, when working towards achievements that pertain to the emptying of your supply bar, you don't need perks helping you regain it.
If you are having trouble emptying your supply bar, you could try the alternate method put forward by Rodeo Legend - use other abilities such as your Molotov to help lower the supply bar, only ensuring that your last pip is spent on resupplying a teammate.