The first bulletresistant soft vest that got widespread use, was a silk-based one built like a gambeson (layers upon layers of silk). King Alfonso the 13th of Spain wore one that saved his life. Interestingly, Archduke Franz Ferdinand of Austria owned one but wasn't wearing it when Gavrilo Princip shot him ... in the neck, though it is likely Gavrilo Princip didn't aim for the neck.
About 1000 AD, the Chinese experimented with paper armor. There were several kinds, but at least one variant was basically a heavy greatcoat-like contraption, and was said to work better when wet. A pistol might have trouble penetrating that - and even a rifle. (Mythbusters did some testing on this.)
It would all depend on the armament of the soldier. A pistol cartridge might not be able to punch through all metal armor - it would depend on the armor and on the cartridge, obviously. Any rifle or light machine gun cartridge should be able to penetrate the metal armor fairly handily.
Depending on the knowledge of the character, he would likely be able to make black powder - but smokeless powder might be beyond him, such as the Poudre B produced by Vielle and later used in the Lebel rifle.
However... scaling things down to 12" from 6' does weird things to stuff like weapons and strength. The square-cube law comes into effect, among other things. His mind might also change because of this; the smallest living (adult) humans have been slightly less than 24", and all of them have had a host of physical problems with their size.
With weapons recoil and bullet weight likewise being scaled down, I doubt that a 150 grain bullet (something like a .303 rifle bullet) scaled down to a 25 grain bullet backed by corresponding less powder would be as effective as one would wish.
Bullet-weight wise you're close to a .22 LR load with something like that, which ends up giving 150-250 Joule of energy to the target under ideal conditions. Medieval armor would have no trouble resisting that - a .22LR is not a toy and can certainly be lethal, but a breastplate will likely stop it (as would leather, lamellar, and so on). And a handgun scaled down like that would be something like an aggressive BB gun - hardly what one might call armor piercing.
He would likely be best used as a sage of some kind - he would be familiar with ways to forge metals, standardization, and so on; remember that the mid-1700s is in the modern period (the American Revolution is after all from 1765 - 1783). He would likely not be big or strong enough to do much more than that - but he might make an excellent watchmaker.