Out of State Sales Tax
Let's say the drop-shipper has nexus in the state where your client is located. In that case you have nexus in that state as well (by extension), and you need to collect sales tax from that client, and then remit it to the state. I am not sure why the drop-shipper would charge you sales tax as well, if the actual end-user transaction takes place between you and the client, not the drop-shipper and yourself (which I assume is a wholesale/reseller transaction)
The drop-shipper might require you to show them proof that you have reseller permit in the states where they have nexus, to ensure you will indeed be collecting and remitting sales tax in those states.
For all tax related questions it's not a bad idea to consult an accountant who specializes in sales tax.