Vernon is likely a character within the Psychonauts universe. In the context of the Vernon Ghost Story, he's probably the central figure who has some sort of encounter or relationship with the ghosts.
Vernon could be anyone. Maybe he's an old man in the neighborhood who loves to share spooky tales from his youth. Or perhaps he's a young person who had a strange encounter and turned it into a ghost story. Without more context, it's hard to say exactly who he is.
Petunia could be a great addition. Her relationship with both Harry and Vernon adds a lot of complexity. She could be the mediator or the one causing more trouble between them, depending on the plot.
In some fanfiction, Uncle Vernon develops by becoming more accepting. For instance, he may start to see Harry not just as a freak but as a person with unique abilities that can be useful. He might begin to tolerate the wizarding world a bit more, especially if he realizes that Harry can use his magic to protect the family from certain threats.
Maybe Vernon is jealous of Harry's magical abilities. He might see Harry as a threat to his 'normal' family image. Also, Vernon has always been a bully at heart and Harry, being an easy target in his house, becomes the victim of his pent - up anger.
Maybe Dumbledore steps in at the last moment and takes Harry away from Vernon's house. He could use his powerful magic to protect Harry and remove him from that hostile environment.
It could be Sirius Black. Sirius is Harry's godfather and he is very protective of Harry. He might break out of his confinement (if he is still in that situation in the fan fiction) and rush to Vernon's house. Sirius has the skills and the determination to face Vernon and save Harry. He could transform into his Animagus form, a large black dog, and intimidate Vernon into letting Harry go.
Perhaps Dudley finally stands up to his father's cruel treatment of Harry. Maybe he realizes that Vernon's behavior towards Harry is wrong and uses his own physical strength or some clever ruse to get Harry out of a difficult situation Vernon has put him in.