Arcane Armor states that it "replaces any missing limbs, functioning identically to a limb it replaces". Would this apply to a missing eye?
An eye is not a limb.
“Limb” is not a special game term in the rules. So, we understand it according to its usual English definition. Since we aren’t talking about tree branches, we must be talking about:
A major appendage of human or animal, used for locomotion (such as an arm, leg or wing).
This is not describing an eye. Prosthetic eyes are a different magic item, in particular, the Ersatz Eye (common):
This artificial eye replaces a real one that was lost or removed. While the ersatz eye is embedded in your eye socket, it can’t be removed by anyone other than you, and you can see through the tiny orb as though it were a normal eye.
Which, if you’re in need of an eyeball, common magic items can be made using the Replicate Magic Item infusion.