How should I approach Toyota or major sponsors to stand out from hundreds of other sponsorship requests?

Instead of coming with your hand out asking what the sponsor can do for you, flip the approach. Come to Toyota and say 'How can I help you sell cars?' Present ideas for how your racing program can help market their core product. Sponsors find it attractive when drivers approach them with solutions rather than just asking for money. Be creative, persistent, and think about what value you can bring to the company rather than what you want from them.