It could be due to a conflict with a particular authority figure at the camp. If he has a major disagreement with a camp leader over how to handle a situation, like dealing with a new enemy. Percy, being headstrong, might decide to run away and handle it his own way instead of following the camp's rules.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
Perhaps he was influenced by some new friends he met who had a different lifestyle or view of the world. They might have convinced him that running away from home was an exciting adventure and a way to break free from the restrictions at home.
He might go to his mother's place. It's a place where he has always felt safe and loved. His mother has always been his anchor, and even though it might be a risky move as monsters could track him there, it could be his first instinct. Another option could be to go to a big city like New York. There are so many people and places to hide. He could blend in with the crowds and use the chaos of the city to stay unnoticed. He could also find refuge in an abandoned building or a place that has some sort of magical protection that he discovers by accident.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
One possible reason could be that he feels misunderstood. In the village, he has always faced discrimination because of the Nine - Tails inside him. People often look at him with fear and suspicion. So, he might run away to find a place where he can be accepted for who he is.
One possible reason could be that Peter might feel like he's a burden to Tony. Maybe he made a big mistake and is afraid of Tony's reaction. For example, if he accidentally endangered the lives of others during a mission while under Tony's supervision.
One possible reason could be that he feels misunderstood. In the Naruto series, Naruto often faced isolation and judgment from the villagers. If this extended to his family in the fanfiction, he might run away to find a place where he is accepted. Maybe he discovers some dark secrets about his family that make him feel unsafe or betrayed.
One reason could be fear. Snape has always been rather intimidating to Harry. Snape's cold demeanor and his past interactions with Harry's father might make Harry feel constantly on edge around him. So, running away could be an instinctual response to this fear.
One possible reason could be to escape the pressure of his family's expectations. The Malfoy family has a long - standing reputation in the wizarding world, and Draco might feel suffocated by the need to uphold it. Another reason could be fear. If he's gotten involved in something dangerous, like with the Death Eaters, he might run away to avoid the consequences.
One reason could be to avoid a dangerous prophecy that they believe the camp can't protect them from. In some fanfictions, they might think that by running away, they can face the threat on their own terms.