I don't know about other countries, but you should certainly not charge tax to customers in the US unless your business is legally subject to taxation in the US states you are delivering to. Here, sales tax is handled by individual states, not the federal government.

Since you are using a US-based drop-shipper, you may well be required to collect and remit sales tax in states where they have a business presence. This is complicated (and dangerous) enough that you should consult with your drop-shipper and get some good US legal advice.