I would like to think that the servers that run ACH networks, IBM mainframes that actually store your account balance, etc are shielded from CME and even an EMP.
But the vulnerable side then becomes the power grid and Internet backbone. Part of a resilient disaster plan would be to allow ACH and other payment traffic to take priority if we were relegated to a low bandwidth connection. If we cannot transfer money, we will collapse, plain as that, once panic over food security sets in.
An EE friend said that we do not have enough transformers to replace if a critical number of them fried. Substation level components could be a huge bottleneck if we had another Carrington event.