Can we sell pet animals / birds online using a ecommerce sit

Have you registered your establishment with any Governmental agencies like Shops & establishment, VAT, Department of animal husbandry etc.?

Established as Shop, Had labour licence. 

What are the activities permitted as per the shops & Establishment license? Also get VAT registration.

Shops & est has nothing to with the nature of business. It is a labour law. VAT is based on applicabilty! So neither of these should bother you.

You may have to get permission from the ANIMAL QUARANTINE AND CERTIFICATION SERVICES (AQCS) - GOI, for permission to custom clearances OR from Department of animal husbandry