Yes, there could be. The Percy Jackson series and the Avengers both have large fan followings, and it's not uncommon for fans to create crossover fanfictions. Writers might be interested in exploring how Percy's demigod abilities would interact with the superpowers of the Avengers.
Well, if Percy Jackson were to become an Avenger in fanfiction, there would be some interesting dynamics. For one, his loyalty to his demigod friends might conflict with the Avengers' way of doing things at times. But his ability to think on his feet, which he developed from all his quests, would help him adapt quickly. And his relationship with the other Avengers, like maybe a friendly rivalry with Thor over who has better control over elemental powers, could be a great plot point.
It could be really cool. Percy might use his water - controlling powers in new and creative ways with the Avengers. For example, he could help Thor with his lightning by creating water - based conductors.
I think Thor would be really tough. Thor has his hammer, Mjolnir, which gives him the power of lightning. Lightning is a very powerful and destructive force. Percy would have to be very careful not to get hit by it. Also, Thor is a god in the Marvel universe, just like Percy is a demigod in his world. So they are kind of on a similar level in terms of their 'godly' status.
In a fanfiction, there are no real limits. Percy could potentially lift Mjolnir if the writer decides to give him the qualities that would make him worthy in Norse terms. Maybe his loyalty to his friends, his courage in the face of danger, and his selfless acts could be seen as worthy traits. However, some fanfictions might also play with the idea that he can't lift it, but still has some sort of connection to it that leads to other interesting plot developments.
He could go to New York City. It has a lot of water sources like the Hudson River where he can use his powers. Also, there are many places to hide from monsters.
He could go to the Underworld of another planet, if such a place exists in the universe he lands in. It might be filled with different kinds of dead souls and ruled by a different kind of Hades - like figure.
You can start by looking on fanfiction.net. It has a large collection of 'Percy Jackson' fanfictions, including many that focus on Percy and Piper. Just use the search bar and type in relevant keywords like 'Percy and Piper' and you'll get a bunch of results.
Annabeth Chase is likely to be one of the supporters. She is smart and strategic, and she can come up with plans to help Percy. Grover Underwood, Percy's satyr friend, might also be there. He can use his nature - related powers to help in the fight. Maybe some of the other demigods from Camp Half - Blood like Jason Grace or Piper McLean could also show up to support Percy.
You can try fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. These platforms usually have a wide variety of fanfictions, including those about Percy and Sally Jackson. Just use their search functions and type in 'Percy Jackson x Sally Jackson' to find relevant stories.
I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide range of wild and creative ideas, but this one seems rather unorthodox considering Artemis' status as a virgin goddess in the original mythology. However, the world of fanfiction is so vast that it's possible someone out there has written such a story.