I had someone point out to me recently that practically all inventory replenishment is now internet dependent. If it were to really go down, there would be far more problems than just your Amazon order or not having movies to watch on Hulu.
Banks, commerce of the sort you mention, food, hospitals, pharmacies, the markets...just think about it.
More seriously, some management of inventory might be possible using computers in offline mode. But the choke point would be the mail systems and public telephone system. Everything would now have to be ordered by letter or by telephone (assuming the latter still worked in degraded form). Since many of these communications systems now control internal functions using Internet-based cloud applications, they would also have to go back to the older paper and analog telephone systems. Ditto for manufacturers, etc. Mass communication would be limited to broadcast television and radio once they too had recovered in degraded form from the loss of the Internet. Assuming no EMP attack to accompany taking down the Internet (a big assumption to be sure), essentially a return to the mid-1980s but with better stand alone computing systems. A lot lost but not everything.
Crippling to be sure. Fatal in time of war? Maybe. With the dramatic reduction in forces heralded by the end of the Cold War, efficient use and support of what remains is vital. Most of that efficiency is based on the rapid communication of information. The Internet facilitates the movement of information but does not generate the information per se. Modern governments do harden their networks against interdiction and disruption. But this is generally limited to military command and control and critical infrastructure. “Disaster recovery” both electronic and physical is part of contingency planning for larger businesses. But how many medium and small businesses have made any such preparations? Probably very few.
In a short war scenario, capability is based on existing forces and their resupply from prewar stocks. The longer the fighting continues, the greater the necessity to generate new forces and refit/refresh existing ones. In short, the emphasis shifts from the adequacy of prewar preparation to sustaining logistics. During a long war, the advantage goes to the side(s) that have the most effective (not necessarily the most efficient) logistics infrastructure. The use of the Internet (or some equivalent publicly accessible wide area network) is a part of that infrastructure. But it is not ALL of what is required. Probably even more critical is manufacturing capacity and transportation infrastructure. Being able to rapidly communicate orders for goods and services that cannot be filled doesn't generate combat capability.
Did you ever hear of schedulers? As long as you have people who know what they are doing, and they have land lines, phone numbers and the ability to pay, then after a couple of days the products will flow.