Shopify can be set up to handle local delivery from either your one or multiple delivery locations. Once enabled the local delivery option would only become an option to those within an owner defined radius, or even defined postcodes, of the delivering store / operation. Local delivery can either be chargeable or set to free.
Customer delivery options and telephone number are entered at checkout.
As with all Shopify transactions a series of email and / or text notifications can be sent to the customer informing them of Order Confirmed; Order Delivered; Order out for Delivery; Missed Delivery
By using the Shopify Local Delivery App (available on iOS or Android) Store owners can optimise delivery routes for whoever is doing the deliveries, stay in touch with customers, and keep track of delivery progress. As an order is delivered is is removed from the list and the next order moves to the top of the list.
See full functionality of the Shopify Delivery app