There are now ubiquitous wireless POS terminals for card payments that can be recharged from emergency sources of electricity(like cars). As long as the mobile internet works it's possible. Of course this only little alleviates the disruption.
If we're talking about a situation where the grid goes down, the mobile internet is most definitely not working.
0:49 and light came back, and woke us for a moment.
So yeah, you need local first POS applications.