This depends on where you are, and where your drop shipper is. For 49 states, this is pretty straightforward: Get registered as a business in your state, and collect sales tax in your state. In California, the process is very convoluted.
If you're in-state, you owe sales tax. If you're out of state, you owe use tax. This tax is based on the state rate + the rate in the county of your customer.
Drop shipping adds a new level of complexity in that if you are out of state, and have a permit, you owe the use tax, but if you are out of state and don't have a permit, the drop shipper owes the use tax. I'm not even sure of the policy if you're in-state.
I'm also still pretty new to this, so I could be wrong. Verify the research yourself, or pay a professional to make sure you don't mess up.
Maybe contact your local Chamber of Commerce? Eve if there is no sales tax, you still might need to register as a business. Are you incorporated/organized in any fashion? Aside from state law, a wholesaler is going to want to know you are buying for resale and that you are a legit business.