Currently reading Matt Fraction's Hawkeye, and I'm even more convinced than I was before that Spider-Man wouldn't work as an Avenger.
Reason I say this is because Fraction's take on Hawkeye is in very much Peter Parker-esque. His Clint is played up as a working-class everyman and the book focuses solely on Clint's side of the MU and not anyone else's (Fraction even starts his run by saying "This is what Clint Barton does when he isn't with the Avengers.").
In the book's defense, it shows that an everyman superhero can be an Avenger and have it feel believable. But it also proves that being an Avenger isn't something trivial, and that it would absolutely affect Spider-Man in ways that no one would not be happy about:
1. When an Avenger gets caught by the police for acting as vigilante and breaking into a mob-owned club to help someone out (something Spider-Man does all the time), he/she can apparently just tell the cops they are an Avenger and to check his/her pockets for an Avengers ID card. Translate this to Spider-Man comics and it would take the punch out of Spider-Man's relationship to the authorities.
2. Clint later gets his membership put on probation by Iron Man for going rogue like that (again, something Spider-Man does all the time) and violating the Avengers' clause. This suggests that an Avenger Spider-Man would have certain rules to abide by even when he is not with the Avengers.
3. The reason New York's mobsters don't want to strike back at Clint is because they're afraid of having a dead Avenger on their hands. At one point they even have a meeting on how to work around this and half the mobsters at that meeting are Spider-Man villains. Surely they would be just as worried of going after Spider-Man if he was a card-carrying Avenger, right?