It actually comes from the old days of musket guns, where a wad (strips of cloth) was packed down in the barrel to create a seal between the gunpowder and the bullet, increasing the internal pressure when the gunpowder fired, and thus speeding the bullet.
A musket is a muzzle-loaded, smoothbore firearm, fired from the shoulder. Muskets were designed for use by infantry.
A soldier armed with a musket had the designation "musketman" or "musketeer".
"I ran out of ammo, General Washington, so I shot my wad."
(When soldiers ran out of bullets, they would resort to "shooting their wad".)
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