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.
Obviously, Harry Vernon is a main character. But often, there are other characters created specifically for the fanfic or borrowed from the original source material if it's related to a known franchise. For example, if it's in a magical - themed fanfic, there could be a wise old wizard who guides Harry Vernon on his journey. Or a young sidekick who accompanies him in his adventures.
The main characters would obviously be Harry and Vernon. Harry, who is in the unusual situation of being Vernon's pet, and Vernon, who has to deal with this new relationship dynamic.
In some fanfictions, Harry might cope by finding solace in his friends at Hogwarts. He could talk to Hermione and Ron about what Vernon does, and their support gives him strength. For example, they might encourage him to focus on his magic studies as an escape from the Dursley's household.
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.
Perhaps in the fanfiction, Vernon discovers a hidden magical talent of his own. Harry, upon finding out, has to decide whether to help Vernon develop it or keep it a secret from the wizarding world as Vernon is a Muggle. This could lead to a lot of comical and tense situations.
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.
One common theme is redemption. Often, Vernon is depicted as having a chance to redeem himself for his past bad treatment of Harry. Another theme is the discovery of common ground. For example, they might find out they both have a love for a certain type of food or a shared fear. And family is also a theme. Despite their differences, they are still family in a way, and fanfictions sometimes explore how that family bond can be mended or redefined.
It might be a fan who was really intrigued by the relationship between Harry and Vernon and decided to take it to an extreme in their writing. There are so many fanfiction writers out there, some write for fun, some to explore different character dynamics. So, it could be anyone really.
Dudley might face his father's anger. Vernon could be furious that Dudley has gone against him to protect Harry.