One possible plot could be that Percy has unknowingly offended Artemis in some way. Maybe he accidentally disrupted one of her hunts or desecrated a sacred place of hers. As a result, Artemis decides to hunt him down. Percy then has to use his wits and powers to avoid her arrows and find a way to make amends.
Percy could try to blend in with mortals. Since Artemis mainly hunts supernatural beings, if Percy could disguise himself well among mortals, he might be able to escape her notice. He could also call on his friends like Annabeth. Annabeth is very clever and might come up with a plan to trick Artemis. Maybe set up false trails or create illusions to mislead the hunters.
In this scenario, Percy would be in a very tough situation. Artemis is a powerful goddess. Percy might have to use all his skills and the help of his friends. He could try to reason with Artemis first. Maybe he'd show her that he's not a threat to her or her hunters. If that doesn't work, he'd probably have to run and hide in places where her powers are limited. For example, in areas protected by other gods who don't want Artemis to overstep.
His friends from Camp Half - Blood could help him. For example, Annabeth. She's smart and could come up with strategies to avoid Artemis. Also, Grover, with his ability to sense danger and his nature - related powers, might be able to lead Percy to safe places.
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.