Hi Aravind, sadly this can only be done with a custom module.
The result could be a domain widget on the shipment method with which you could freely configure that the shipment method only appears if the sales order has a customer which is a paying member and the membership type is of type X.
Very technical: to do this, I would adjust the method _match_address() in the file odoo/addons/delivery/models/delivery_carrier.py to add extra matching conditions
We'ld be very happy to help, just contact us via hi@latus.ch. Good luck and thanks!