Palm pistol doesn't show up as eligible for the Artificer Infusion Repeating Shot

That’s because it is not eligible as it does not have the ammunition property as per the author, Matt Mercer’s wishes.

It's details says it does, but it doesn't have the tag. There's a mismatch:

https://www.dndbeyond.com/equipment?filter-search=Palm Pistol&filter-cost-min=&filter-cost-max=&filter-weight-min=&filter-weight-max=

No mismatch. That was intentional design. (I suspect it was specifically intended to prevent the infusion from being possible.)

It's important to note that Matt Mercer's firearms don't use the same ammunition rules as what is described in the rules of D&D.

Here is the ammunition property rules 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.

And here is Mercer's ammunition rules (which are not a property, just a rule)

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.

Because of the rules mismatch between what D&D uses and what Mercer designed, the Exandrian firearms don't have the ammunition property and are thus incompatible with repeating shot.

Specifically, the gunslinger firearms use ammunition, but don't have the ammunition property. This is by design since the ammunition property does not mesh with extra attack without an otherwise unrelated feat.