If you are working with just ONE drop shipper, there is a solution (I believe) that is adequate, but it may depend on a little "manual processing" if you are not using an online (realtime) payment processor or gateway.

Firstly, are you working with just ONE drop-shipper? And, is EVERY product in your store handled by that drop-shipper?

If YES to the above, then simply add that drop-shipper's e-mail address to the list of recipients in ADMIN>>>CONFIGURATION>>>EMAIL OPTIONS>>> Send Copy of Order Confirmation Emails To...

If you are taking payment real-time (credit card gateway), then you will have been paid by the time the email is sent to your drop-shipper.

Currently, there is no way that a standard zencart installation can "split up" an order confirmation e-mail (ie: determine which drop-shipper gets what elements of a particular order, and just e-mail them those parts).

If you are working with several (more than one) drop shipper, the process has to be manual.