The Avengers might be initially shocked. For example, Iron Man could start by using his technology to find ways to help Hawkeye communicate better within the team. He might create special devices for him.
They would need to make some adjustments. For communication during missions, they might rely more on visual cues. Hawkeye, with his excellent archery skills, could use his own set of signals to indicate things to the team.
Another option is Wattpad. There are some great fan - created stories there as well. Some authors on Wattpad like to put their own unique spin on the Black Widow - Hawkeye dynamic, such as setting their adventures in different alternate universes which can be really fun to read.
On the other hand, if some of the fanfiction was a bit too outlandish or inappropriate, it could make things a bit awkward. Hawkeye might feel embarrassed, and the other Avengers might not know how to react. But they'd probably try to smooth things over and remind Hawkeye that it's just the creativity of the fans and not a true representation. They'd still support him as a teammate.
Tony Stark would likely find it hilarious. He'd probably be the first one to start making jokes about it. He might say 'Oh, Hawkeye, you've got some really creative fans out there. I should hire them to write my memoirs!'. As for Hulk, he probably wouldn't care much. He might just grunt and go about his business. But if he did read some, he might not understand it very well and get frustrated. Hawkeye himself would probably be embarrassed at first, but then he'd start to see the humor in it and might even enjoy some of the more positive fanfiction about him.