Fury even gives Peter a new Stealth Suit so that nobody knows that Spider-Man has left his neighborhood.
We don’t know why they need him specifically, but it must be something that requires his skillset and can’t be achieved with their own guns and spycraft.
This image from Reddit shows that Nick Fury and Maria Hill are very much on the job and counting on Spider-Man to execute some important task towards uncovering the mystery of the Elementals and stopping their attacks.