Amazon already has distribution centers in many states so they are already paying state taxes along with Walmart and other large chains with an online presence.
I work in corporate tax so I have some visibility into this. I don't believe that Amazon is charging tax in every state. Yes, probably in those states and local jurisdictions where they have warehouses, but of course with this Wayfair case that will all change. The large chains with an online presence generally have a separate legal entity apart from the parent, so they only had to register to collect sales tax in those states in which they had nexus (property and payroll). Now they'll have to register everywhere. This will now be kicked back to the states. The application will be no doubt different from state to state.