Percy would probably be really confused at first. I mean, one moment he's a normal demigod and the next he's a jackalope. He'd likely try to figure out what happened and how to change back right away. He might be a bit frustrated with Artemis for doing this to him without much warning.
The fanfiction could start with Artemis getting angry with Percy for some reason, like he disrupted one of her hunts. So she uses her powers to turn him into a jackalope. As a jackalope, Percy is small and vulnerable, but he also has some unique skills. He can jump really high, which he uses to escape from predators. His friends are shocked when they see him, but they quickly decide to help him. They go on a journey to find Artemis and ask her to turn him back. On the journey, Percy has to deal with being hunted by other creatures who see him as an easy target, and he also has to learn to control his new body.
Percy might initially be in shock. But being a resilient character, he would start by observing how girls around him act. He'd probably learn from the female friends he already has in the Percy Jackson universe. Maybe he'd also rely on his natural instincts and the courage he had as a boy to face the new challenges that come with being a girl, like different social expectations and physical changes.
Perhaps Artemis is testing Percy in some way. By turning him into a girl, she wants to see if he can still be as brave and resourceful. The plot could involve Percy learning about the female experience among the demigods. He could discover new aspects of friendship with female characters who now see him as one of their own. And he has to figure out how to reverse the transformation while also dealing with the current challenges.
In his family relationships, Sally Jackson would be in for a big surprise. She would have to adjust to seeing her son as a daughter. Percy would have to deal with different family expectations as a girl. His relationship with Poseidon would also be different. Poseidon might be confused and perhaps less likely to immediately offer the same kind of support he did when Percy was a boy. And Percy's romantic interests, if any, would have to be re - evaluated. He might find himself attracted to different people or be the object of different types of attraction.
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.