Upvote:2
The claim by China Air that they don't have any passenger names sounds like a rather non-standard agreement. Normally a space is booked in the passenger's name on the operating carrier (CI), when you make the booking with the marketing carrier (VN).
I would ask Priceline (as they are the agents) for the PNR for your booking(s). There should ultimately be two PNRs (one for Vietnam and one for China) though you may need to ask Vietnam for the CI PNR directly. China Air maybe able to locate your booking armed with their PNR as a reference.
But there is also the possibility that the discounted airfare you purchased does not allow advance seat selection. And if this is the case, you best keep your fingers crossed that you will get lucky come the 24 hour mark.