Step outside your comfort zone!
We all have local businesses around us where delivery is not an option. For example, maybe you have a local cupcake shop in town that is pretty popular. Try to set up an appointment to meet with the owner to discuss if they would like to help advertise your business in exchange for you advertising that you deliver for them. WIN WIN! for both businesses. Pick a local store, or place that you would like to possibly deliver from on a regular basis, and send them a message to their FB business page.
For example... Hello, my name is (Your name), and I own and operate a local delivery service here in (city) called (Business name). I was hoping to set up an appointment with the owner. I am interested in possibly advertising delivery from your store. In exchange, I would like the ability to leave some business cards for your clients to have access to. If it would be something the owner is interested in please have them give me a call (phone number), or let me know an appropriate time I could come down to discuss it. I appreciate your time and hope to hear back soon. Thank-you!