Arm Blades are Magic weapons, infusions can only be applied to non-magic items.
Even if you ignore that:
Arm Blades are Melee weapons, Melee weapons don't have the ammunition property, you would need the Returning Weapon infusion.
However, there used to be an item called an Armbow, which does almost exactly the thing that your player is describing. Having a custom infusion that acts Armblade but works with one-handed ranged weapons instead seems reasonable. Giving it the unlimited ammo power but not the +1 bonus of Repeating Shot should keep it more or less inline with other infusion options, including the normal Repeating shot and the Wand Sheath.