Start by contacting a local shipper or a local franchisee of a larger shipping company. Ask if they might like to partner with you.

Another option would be to contact an individual donor and ask that person (or company) to exclusively sponsor this portion of your operation. In turn, you can name the service after the sponsor/donor, and use that to leverage any additional promotional materials or public relations you may send out or produce to get or solicit additional donations.

When you contact these people, think in terms of how you would sell something. By this, I mean you should begin with a list of your target shippers/donors/sponsors and then create an offer that will benefit them. Tell them that by partnering with you they can be part of a great effort to help others, and be recognized in the community for doing it.
Then you’ll need craft your pitch or message from there.

You'll also need to make sure you have all licensing and other necessary paperwork in place in terms of what is required for a not-for-profit in your area. Your partners and donors will want to know they are dealing with a credible and legitimate program before they extend help or resources in your direction.

Related: How to Start a Nonprofit
Related: Seven Rules for Nonprofit Fundraising Success