Start from the inner circle, which is the kid's families. With a presentation leaded parent meeting can initiate. It is better, if the kids conduct the presentation. This engages parents very much.
This project is a good opportunity for the school to reach out it's community. Administration can join the adventure of hunting sponsors by advertising the projects in the school?
A website can really help to cover everything needed, when you are in a room with your potential sponsors. Team members can show the website, which has:
an itemized list of materials and costs,
pictures of team and some work if done,