Paladins don't have to serve gods, in Lucia's home plane. An oath is enough. For the first few flickerings of power, not even the oath is required - only the sort of conviction that leads people into taking oaths once they realise what they are becoming.
It's still very, very helpful to be in some sort of order, because there are a lot of evil forces in the world that would rather kill baby paladins before they become adult-paladin-sized problems, and because it is genuinely very difficult to keep to the straight and narrow path of the light when you don't have other people around you supporting you and pushing you to become better, and also because maintaining armour is really expensive and paladin orders have people who can help with that sort of thing.
It is probably her own fault that she is not in an order and she doesn't have armour and she barely even has the rusted sword that she clings to while she climbs the long, long weary miles up this mountain to the tiny nameless shrine that is supposed to be at the top. The villagers said they didn't know the name of the god that lived up there and didn't know what they were the god of. That's okay. They didn't know her name either, but they gave her advice and shelter because they could see that she was good, and they pray to the god whose shrine is atop their mountain because the god is good.
And Lucia is desperate.