Yes, eBay has its own courier services. named - eBay global services. But this service is not yet applicable in India. if you are an eBay seller from the country which provides this service. Then you just need to Login in your eBay global services and then you can easily use this facility provided by eBay.
eBay India delivers its order to its customers' through these logistics partners-
• India Post
• FedEx Express
• Aramax delivery unlimited.