Maybe in the fanfiction, there's a secret organization that the Avengers and Spider - Man both discover. They have to team up to take it down. Spider - Man's street - level intelligence and the Avengers' global - scale resources combine really well. This successful teamwork makes the Avengers think he should be part of the team right away.
It could be written that a common threat forces Spider - Man and the Avengers to work together. During this cooperation, the Avengers see his skills, his heart for justice, and his ability to think on his feet. They then decide to extend an invitation to him to become a full - fledged member early on.
In some fanfictions, it could start with Spider - Man accidentally getting involved in an Avengers - level threat. He manages to hold his own and even helps them out a bit. Impressed by his performance, the Avengers then start to consider recruiting him. They might send one of their members, like Iron Man, to have a talk with Spider - Man, highlighting the benefits of joining the team, such as access to better resources and more global protection opportunities.
Well, one thing that could happen is that the team dynamics would change a great deal earlier. Spider - Man's youthful energy and unique powers, like his web - slinging and spider - sense, would add a new flavor to the Avengers' combat strategies. He could also form different relationships with the members sooner. For example, he might have an earlier mentorship with Iron Man or a friendly rivalry with Captain America.
Another possibility is that the Avengers are being manipulated by a villain. The bad guy could be using false information to make the Avengers believe that Spider - Man is a danger. So they go after him without really knowing the truth.
The plot might involve Spider - Man getting into some trouble in New York and the three Avengers coming to his aid. It could be that a powerful alien artifact has been stolen and Spider - Man accidentally gets blamed for it. The Avengers then step in to help clear his name and find the real culprit while also protecting the city from the potential danger of the artifact.
In the fanfiction, it could be that Spider Man has a personal vendetta against one of the Avengers members. For example, if one of them had a past with a villain that Spider Man has a strong connection to, like if an Avenger unknowingly helped the villain in the past, Spider Man might transfer his hatred for that villain to the Avenger and by extension, the whole team.
He could be injured by a new and powerful villain that the Avengers and Spider - Man are facing together. The villain might have a weapon or power that can penetrate Spider - Man's suit and defenses, like a sonic blast that disrupts his spider - sense and then physically harms him.
Spider - Man could use his agility and web - slinging skills to escape. He could swing between buildings, making it difficult for the Avengers to follow him. His small size compared to some of the Avengers also gives him an advantage in hiding in tight spaces.
One reason could be a misunderstanding. Maybe some events were misinterpreted, and the Avengers thought Spider - Man was involved in something bad. For example, if there were some crimes that seemed to have Spider - Man's fingerprints all over them, but in reality, he was being framed.
Spider - Man would likely be confused at first. He respects the Avengers and can't understand why they are suddenly after him. He might try to talk to them, to explain that he's innocent. For example, he could swing up to Captain America and start pleading his case.
Spider - Man would probably be shocked at first. He looks up to the Avengers and being kidnapped by them would be the last thing he expects. He'd likely try to talk his way out, using his quick wit and charm. He might start by asking why they're doing this and try to convince them that they've made a mistake.