It could be to spite Voldemort. Snape was secretly working against Voldemort. By saving Harry, he is ensuring that Voldemort's plan to kill the boy fails, which is a big blow to the Dark Lord. Also, Snape might have a sense of justice deep down. Despite his often - prickly exterior, he doesn't want an innocent baby to be harmed by someone like Vernon. And he may see a bit of himself in Harry, a child who is at the mercy of a cruel guardian, and that could be another reason for him to step in and save Harry.
Snape might do it out of his love for Lily. Since Harry is Lily's son, he feels a sense of obligation to protect him.
In this type of fanfiction, Snape could simply intimidate Vernon. Snape has an air of mystery and a certain power about him. He might show up at Vernon's doorstep, give Vernon a cold stare, and with his reputation as a powerful wizard, Vernon might be too scared to oppose him. Snape could then just take Harry without much resistance from Vernon.
One possible reason could be that Snape still has an underlying sense of loyalty to Dumbledore. Since Dumbledore cares about Harry, Snape might step in to save him from Umbridge's cruelty. He doesn't want to see Hogwarts turned into a place of unjust persecution under Umbridge's regime, and Harry is a symbol of the resistance against that.
One reason could be his sense of duty. Despite his dislike for Harry on the surface, he might still feel responsible for the safety of all students at Hogwarts, including Harry. So when he sees Harry in danger from Dementors, he steps in to save him.
In this fan fiction scenario, Snape might use his knowledge of the Muggle world and Vernon's weaknesses. Snape could have found out some information about Vernon's business affairs or personal secrets. Then he blackmails Vernon into leaving Harry alone. By doing this, he effectively saves Harry from Vernon without having to use too much magic and draw unwanted attention.
It could be that Snape wants to spite Vernon. Vernon represents everything that Snape despises about Muggles who are cruel and unaccepting. By saving Harry from Vernon, Snape is showing his superiority over Vernon and also his power as a wizard. And of course, there's still that underlying reason of his love for Lily which makes him want to protect her son.
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.
One possible reason could be that Harry is afraid of Snape's intimidating presence. Snape has always been strict and rather unfriendly towards Harry in the original series, so in the fanfiction, this fear might be amplified, causing Harry to run. Another reason could be that Harry has stumbled upon something he shouldn't have seen that involves Snape, and he's running to avoid getting caught.
Vernon might be jealous of Harry's magical abilities. He could see Harry as a threat to his 'normal' family life and his authority in the household.
One possible reason could be that Snape sees Hermione's value as a brilliant student. He might recognize her potential and not want it to be wasted just because of her dad's actions or situation. Maybe he has a sense of justice deep down, despite his often cold exterior.
One possible reason could be Snape's lingering resentment towards James Potter, Harry's father. Snape might see Harry as a reminder of the bullying he endured from James during their school days. So, he takes out his anger on Harry.