Perhaps they'll be so into the fanfictions that they'll try to incorporate some of the fictional elements into their real lives. For example, if there's a fanfiction where Tony has a unique way of powering up his suit, they could try to replicate it in a way that's applicable to their own tech. It could also strengthen their bond with Tony as they get more insights into his character through these fictional works.
First of all, the Avengers realizing they need Tony fanfiction could be a sign that they are missing him deeply. They might read these fanfictions to relive the memories of their time with Tony. In the fanfictions, they could see different versions of Tony's relationships with each of them. It could make them appreciate their friendship with him more. Also, the fanfictions could offer new perspectives on Tony's genius, which might help them in their future endeavors. They could even share these fanfictions among themselves and have discussions about how the stories could have been different in real life.