Arizona's online tax is a relatively new animal, and seems to be in additional to the normal sales tax regime. The practical reality here is your state will be using this law to go after deep pockets like Amazon or Ebay. Small retailers like your supplier in TX are probably going to fly below the radar, and if you have to phone your order the argument that its not an online sale/store is pretty solid IMO.
I can't speak to the new online tax law there, but for traditional sales taxes a state cannot impose a collection responsibility on a vendor who has zero physical presence in their state.
Edit: This new AZ law is not a separate tax regime, but is changing state law to conform with the Supreme Court case South Dakota v. Wayfair
https://www.aicpa.org/advocacy/state/south-dakota-v-wayfair.html. Basically overturns the physical presence requirement I mentioned earlier for large retailers (over 200 transactions or $100,000 or sales to state residents)