This should not be a problem
When has that ever happened? Sure, payment processing systems have crashed or have slowed down at retailers in the past however, it would only be an issue if Amex, Mastercard, Visa, Discover were unable to process the transactions. Those systems do not use the internet to talk to each other. The use systems such a Plus, Most, Accel, CU 24 and Co-Op to name a few.
Summer of 2016 in my city/county. A squirrel fried a main transformer and it took anywhere from 6-11 days to restore power, depending on where you were located. The banks closed b/c w/o power there are no ATMs, vault doesn't open, etc. Gas Stations accepted cash only. Grocery Stores were closed and all freezer stock spoiled. No air conditioning, no lights, and all fresh and frozen food spoiled in residences. Thank heavens I keep $500 in cash for emergencies and had some emergency food. Of course I was able to charge my cell phone by starting up the car (and enjoying the blessed air conditioning for 30 minutes once a day). It really was a reminder that any interruption to the electrical grid can put an entire town and surrounding area back to a very primitive time.