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.
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.
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 possible aspect to analyze is Snape's complex character. In many 'Snape Saves Harry' fanfics, it delves deeper into his motives. Snape has always had a love - hate relationship with Harry. In these fanfics, his act of saving Harry might be due to his lingering love for Lily. It shows a different side of Snape, one that is more heroic and self - sacrificing than what we see in the original series at times. This also allows for exploring different plotlines and character development arcs for both Snape and Harry.
I also like 'When Snape Becomes Harry's Guardian'. In this fanfic, Snape saves Harry from a dangerous situation early on, and then takes on a sort of guardianship role. It has some really interesting interactions between the two characters as they learn to live with each other.
You can start by looking on popular fanfiction websites like FanFiction.net or Archive of Our Own (AO3). Just use the search function and type in 'Harry Saves Vernon' and you should get a list of relevant fanfics.
In the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.
After saving Harry, Snape may have to deal with a lot of mistrust from some people in Hogwarts who don't understand his motives. However, he might also start to form a closer bond with Harry, which could change the whole dynamic between them. This new relationship could lead to some unexpected alliances and enmities within the wizarding world. And if the Dursleys were to try and retaliate, Snape would have to constantly be on guard to protect Harry from any potential harm they could cause.
The Dursleys might be left in a state of shock and confusion. Since they have been so mean to Harry, they could be worried that the wizarding world will come after them for revenge. But since they are Muggles, the wizarding world may just leave them be, but they will always live in fear of the unknown.