One idea could be that Artemis becomes mortal to understand Percy's human world better. Maybe she's intrigued by his heroism and wants to experience life from his perspective. This could lead to a whole new set of adventures as she struggles with mortal limitations.
You can try Archive of Our Own (AO3). It has a large collection of fanfictions, including many about Percy and Artemis in the mortal context. Just use the search function and type in relevant keywords.
There could be a love story. Artemis, now mortal, might develop feelings for Percy that she couldn't as a goddess. This would be a complex relationship as Percy already has his own life and relationships. The fanfiction could explore how they navigate these new emotions.
Well, there's 'Percy's Mortal Adventure with Artemis'. This fanfiction takes the idea of Percy being mortal and throws him into a series of challenges with Artemis. It's full of action and the relationship between the two characters is explored in a unique way. There are some really cool fight scenes and the story keeps you on your toes throughout.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.
Well, Artemis is a complex character herself. Perhaps she sees something of herself in Percy, like his determination or his sense of loyalty. Fanfiction often takes these qualities and twists them in new and interesting ways. She might be drawn to how Percy is portrayed in different fan - written stories, whether it's his heroic deeds or his more vulnerable moments. Reading Percy fanfiction could be a form of escapism for her, a way to explore a different kind of heroism outside of her own experiences.