Yes, you can start it as a home-based operation as long as you comply with local and state regulations--or the rules for your particular area. Non-profit status is something else you will need to research with a competent attorney or CPA as there are federal rules and regulations that govern non-profit status.

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.