As far a business sponsorships go, you're answering the "why should I pay you?" question. Be concrete and confident (quantitative can help too). What does your team do for students, aka future professionals and citizens? For the community? How will the
sponsor's support help that? What will they get out of it?
Tailor your message and your presentation to the audience. Is it a multi-national STEM enterprise that's looking for a presentation and demo? Or a 2-person company one of your mentors collaborates with, which might want a shop tour? Parents and mentors do make great feet-in-the-door. Or a multi-national STEM enterprise that's looking for a presentation and demo? Note that some of those larger companies and organizations may have grant programs, which can also prove beneficial.
It can be hard, especially in this economy, to stir up interest. Make your presence felt in the school and community, and remember the answer is always no if you don't
ask. It can be hard, but it's a lot easier than trying to pull money out of thin air.
Team info packets can also help get you noticed.
Remember that there's also community events/fundraisers, though. Bake sales, candy vending,
pancake breakfasts, and
pasta dinners can have pretty big nets. Even non-food related fundraisers can rather productive.
Our team hosts an annual VEX event to the tune solid profit.
Overall, stay organized about it. You don't need to become a 501(c)(3) right off the bat--though finding a cooperative one may help--but stay on top of your budget and bank account. Be clear about what money you need and have and where it's going. If you can, find mentors with experience/expertise in the area. Get people involved! Students, parents, mentors, robots, cousins, dogs. The more the merrier (and usually, easier).
If I've been unclear, FIRST NEMO is also a fantastic resource. Good luck!