How can I start a non-profit food bank in my neighborhood?
Learn how to invest your IRA or 401k into a franchise penalty-free. ($50k min)
That said, you could also look to create a for-profit business, in which you add a small fee to the overall total for your delivery service. You can use your fee or whatever profit you make to buy additional food or create more routes for your customers, and you may be able to partner with a delivery operation that is currently servicing routes and create a joint venture with that company to deliver food.
You could also look to partner with a current non-profit and offer delivery services, and work out a fee schedule where some patrons pay and some of the more needy don't.
Bottom-line? You have a lot of options, just make certain you follow the legal standards for either a non-profit or for-profit operation, as well as any food handling statutes you may need to follow.