I have a bit more detail after further troubleshooting
Link to the store: https://www.chemsw.com/Software-and-Solutions/ChemSW-Store/
Shipping field updates always seem to fail when checking out with a cleared browser cache and history. The page reloads when a shiping options is selected, but the shipping value does not update.
After other activity however, (e.g. deleting an item, adding an additional item, returning to the store then back to the cart, etc) the shipping field seems to behave exactly as expected. The page refreshes when a selection is made and the total updates.
Once it is working, it contiues to work just fine until I clear my cache and my cookies, at which point it again validates, but fails to update the total.