Well, some 'Avengers sick fanfiction' might explore the emotional side of a sick Avenger. Imagine Thor getting sick and feeling weak for the first time in a long while. He might struggle with his self - identity as a strong warrior while the other Avengers, like Bruce Banner, use their scientific knowledge to figure out what's wrong. There are also fictions where Clint Barton gets injured and sick due to a battle, and the team has to deal with the fact that he can't be as active in the field as usual.
There are 'Avengers sick fanfiction' that take a different approach. For instance, in some stories, it's not a physical illness but a kind of mental or magical 'sickness'. Maybe Wanda Maximoff accidentally uses her powers in a wrong way and it makes her 'sick' in a sense that she loses control of her emotions and powers. The Avengers then have to help her regain her balance, which could involve some really interesting character interactions and plot developments.
One popular 'Avengers sick fanfiction' might be those that focus on a character getting sick during a mission and how the team rallies around them. For example, there could be a story where Tony Stark gets a strange alien illness and the rest of the Avengers have to find a cure while also protecting the world from threats. Another could be about Steve Rogers getting a common Earth illness and the team's humorous attempts to take care of him, like Natasha trying to cook soup but being really bad at it.