Though Spider-Man has been a member of the Avengers in the past, the web-slinger originally rejected the Earth's Mightiest Heroes. When Spider-Man was first introduced in 1962, he was originally a loner who was considered a hero by some, and a menace by others.
Though Spider-Man would ally himself with various super-heroes over the years, including the Fantastic Four, The X-Men, and Daredevil, he never formally joined any team for a long time. Because of Spider-Man's dubious reputation, despite his numerous acts of heroism, no one attempted to give Spidey a fair shot at really belonging to the hero community. However, that would change a few years into Spider-Man's heroic career, when the Avengers decided to invite the web-head to join their team!
In Amazing Spider-Man Annual #3, by Stan Lee, John Romita, and Don Heck the Avengers debated on whether or not to invite Spider-Man to their team. Members like Hawkeye, who had been considered outlaws, were more than willing to give Spider-Man a shot, while others like the Wasp had decided against the idea due to the mystery around the hero. While the Avengers could not reach a consensus at first, Captain America decided to reach out to fellow hero, and Spider-Man's friend, Daredevil to see if Spidey would be considered Avengers material. After Daredevil gave Cap a glowing recommendation, the team decided to track Spidey down and formally invite him to join.
After a short time, Thor managed to track down Spider-Man and invited him to try out for the Avengers, but Spidey was initially reluctant, taken aback by how abrupt the invitation was. Thor gave Spider-Man 24 hours to officially reply, and though Spider-Man spent some time thinking the issue over, he ultimately decided that the best use of his powers was to join the greatest super-hero team. The Avengers welcomed Spider-Man with open arms, but before he could officially join, they gave him the ultimate test: Find and Capture the Hulk!
Spider-Man, with very little hesitation decided to track the Hulk down, and promptly began searching for him in New York. After swinging by the Daily Bugle, much to the annoyance of J. Jonah Jameson, Spidey learned that the Hulk was spotted at a Gamma Ray Research Center. Once Spidey found Hulk, the two began to fight and, despite the Hulk's strength, Spider-Man was able to keep up with the Green Goliath. At one point, Hulk gets doused with Gamma Rays, causing him to revert to Bruce Banner. Spider-Man and Bruce converse, with Spider-Man asking why the Avengers would want to find him. Bruce relays how he is constantly hunted due to his transformations, and he has no idea why the Avengers want to find him.
Banner eventually reverts back to the Hulk, and though Spidey manages to subdue him in webbing, he feels that Bruce and the Hulk should not suffer anymore by being captured. Spider-Man lets the Hulk go, and returns to the Avengers by flatly saying that he has failed his mission. After Spider-Man swings away, the team are disappointed at the hero's apparent inaction, and discuss how they only wanted to help cure the Hulk, rather than imprison him (a detail they failed to relay to Spider-Man before he left).
In the aftermath of Spider-Man's failure, the hero laments the opportunity that could have been, now longing for a membership he was initially reluctant to have. This contrasts with Spider-Man's portrayal in Spider-Man Homecoming, where Spider-Man also rejected the Avengers, albeit for a different reason. In Homecoming, Peter Parker had idolized the Avengers for years, and would have done anything to join the super-team. By the end of the movie however, after defeating the Vulture and learning what it really means to be a hero, Peter decides that he's fine with being a "friendly neighborhood" Spider-Man for awhile. In the comics however, Spider-Man never had any interest in joining the Avengers, preferring to call his own shots and be a hero on his own. While Spider-Man soon warmed up to the idea of joining the team, he didn't want to turn in the Hulk to the authorities, having seen that his life is downright miserable. While the Avengers only wanted to help their ally, Spider-Man wasn't privy to that fact, and thought that he did the right thing by rejecting his membership. In the movies, Spider-Man happily turns down his membership to enjoy being young for awhile, while in the comics, Spider-Man ends up more depressed by the whole experience.
Though Spider-Man's initial encounter with the Avengers didn't end well, he would eventually become a reserve member of the Avengers, fighting alongside them from time to time. Eventually in Brian Michael Bendis's New Avengers, he would become an official member of the Avengers alongside other heroes like Wolverine, Luke Cage, and Spider-Woman. For several years afterwards, Spider-Man was a member of several Avengers teams, including the main Avengers, the New Avengers, and even the Uncanny Avengers for a brief time.
While Spider-Man is not currently a member of the Avengers at this time, it's only a matter of time before Spider-Man answers the call to action and becomes a Mighty Avenger once more!