The Artificer Infusions that allow you to Infuse with a weapon, or armour, will not work with Exandria firearms. When you click on the Infusion section in the "Inventory", and go to Infuse a Exandria weapon with "Repeating Shot" it doesn't recognise that I even have Exandria weapons in my inventory.

Repeating Shot works with weapons with the ammunition property. Exandria  firearms don’t have the ammunition property. 

Sorry for the necroposting, so explain me how does the Exandria weapons works, it shoots bullets from thin air?

Of course they require ammunitions, it’s because it’s replaced from the "Reload" one, but what does it exactly reload though?

Y’all guessed it right: ammunitions.

Why don’t make the Infusion take in account weapons with the "Reload" properties too? Or it is too much hardcoded to make that possible?

Because RAW the infusion requires that a weapon have the Ammunition property. Those weapons may use ammunition, but that’s not the same thing as having the Ammunition property.

This is the Ammunition property:

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.

This is what the Exandrian weapons say about ammunition:

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.

See how they’re different?