Not sure if this is intended or not, but it is not possible to choose an Exandrian firearm for Artificer Infusions such as Repeating Shot if you are both a Gunslinger & Artificer.

The Repeating Shot infusion requires a martial weapon with the ammunition property.  Exandrian firearms do not have the ammunition property.  Therefore they are not eligible for Repeating Shot.

Just to add to ArwensDaughter’s post:

The Ammunition property is a very specific rule in D&D, not just something that requires ammunition:

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.

Exandrian firearms require ammo, but that’s an entirely different 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.

 
I hope that helps.