Brink

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.

It's required that you are a soldier for this achievement.

The description states that you must give away the last of your supplies to refill a teammates ammo, and ammo refill is exactly what the X botton does when you are a soldier.

You could do this a couple ways.

1. Run around giving ammo by pushing X to your teammates untill all your section on the supply meter have run dry.

2. Use other abilites like molotov cocktails until your supply meter is down to one bar, then, using the last bar, give a supply of ammo to a near by teammate.

Note: The supply meter appears in the lower left corner and is devided into white bars that circle around the icon that indicates what you have set to which botton on the D-pad.