Not at all. Implicit in the reasoning that products sold on the Internet are free of state sales taxes is that they are sold in one state and shipped to another. Neither state -- the one where the order is packaged nor the one where the product is received -- has the legal jurisdiction to charge a sales tax.
This is all slowly disappearing anyway. States with sales taxes have laws that define what is a "domestic" transaction (subject to sales tax) and what is an "interstate" transaction (where a sales tax would be prohibited under Federal law), and the distinction usually involves the question about whether the retailer has a physical presence in the state. Amazon is opening distribution centers in so many states nowadays that their shipments are meeting the definition of a "domestic" transaction almost everywhere they operate.
You are correct regarding "sales" tax, but most, if not all, states with a sales tax also have an accompanying use tax for this very purpose. That use tax would apply to those things shipped to the consumer from out of state and used/consumed by the consumer. Under the use tax, the consumer is the one responsible for paying the tax to the state, not the seller.
I’m always shocked we felt the need to reinvent this wheel for internet commerce. My family was in mail order in the 70s and 80s, the laws were well defined and easily followed: if the company had a physical presence in the sales tax zone sales tax had to be charged, if no the buyer is supposed to self confess on their income taxes. It was right there on the mail order form “residents of blah, blah, blah add x% tax”. Internet commerce IS mail order. All the same rules should have applied, but somehow the world got snowed under by the .com boom and thought it was a brand new thing needing new rules.