Your Call Center is where your CSR's take telephone orders and your dispatchers dispatch your drivers. While some smaller services can operate out of the owner's house, most services will need to lease some office space. It doesn't have to be fancy, or located in a high-rent section of town. Being a delivry services, your customers will not be coming to you; you will be going to them.
You office will need teh standard office equipment of a fax machine, several telephone lines (depending on your volumne), a copy machine or all-in-one printer, and several computers. Depending on your software setup, you may need a server or two as well.
A high-speed Internet connection is mandatory these days, whether you host your own site or not.
You call center will need ot be open whenever your services is, and usually a little before and after as well. So if you are planning to run it out of your house, keep that in mind.