It seems that all of the gunslinger firearms added do not have the ammunition (firearm) property. This seems to be a small oversight, but it does mean that, as an artificer character, you cannot use the "repeating shot" infusion on a gunslinger firearm as they don't have the correct property.

I have attempted to make a custom magic item and add this property so that it will work, but the ammunition weapon property doesn't show.

This is correct per Matt Mercer's original and current design of the Gunslinger. None of the Exandrian firearms have the ammunition property.

Except that they do in the item description.

They have Matt's own ammunition description, not the ammunition property as defined in the rules of D&D. 

Here is the description of ammunition taken from 'Bad News'

Ammunition
All firearms require ammunition to make an attack, and due to their rare nature, ammunition may be near impossible to find or purchase. However, if materials are gathered, you can craft ammunition yourself using your Tinker’s Tools at half the cost. Each firearm uses its own unique ammunition and is generally sold or crafted in batches listed below next to the price.

However, here is the description of the ammunition property taken from the basic rules:

Ammunition. You can use a weapon that has the ammunition property to make a ranged attack only if you have ammunition to fire from the weapon. Each time you attack with the weapon, you expend one piece of ammunition. Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon). At the end of the battle, you can recover half your expended ammunition by taking a minute to search the battlefield.

If you use a weapon that has the ammunition property to make a melee attack, you treat the weapon as an improvised weapon (see "Improvised Weapons" later in the section). A sling must be loaded to deal any damage when used in this way.

Ammunition as Matt uses is not the same as the way D&D uses it; none of his guns have the property that exists in the rules. You can see this in the PDF of the class as it exists on the DMs Guild:

Also, if you look at that pdf, it lists ammunition as a description, not a property like 'misfire' or 'explosive'

Right, which means that it's an oversight not including it in the properties.

At the very least the option to give a weapon the "ammunition" weapon property should exist for the homebrew content, which it does not.