The WP eStore wasn’t designed for a drop shipping type usage so it doesn’t have a built in function for this.
However, most drop shipping companies have a method whereby they accept the payment notification (IPN) to be sent to one of their API/Script and they can do the drop shipping for that transaction.
WP eStore has a feature that allows you to send the payment notification (IPN) to a 3rd party URL or script. Simply use that feature and eStore will forward the instant payment notification (IPN) to this 3rd party script everytime a payment for a eStore product is completed.
Check the following settings:
WP eStore Settings -> 3rd Party Integration -> POST IPN to a 3rd Party Application