Sure banks can. Source: did software security for a number of banks. Big banks are chock full o' CSRFs, XSS, SSRFs, and SQLIs. They get found all the time. For every valid report they get, they get 3 that aren't valid. Nobody's hair achieves ignition over this stuff.
There are two types of financial service organizations: the big banks, and random firms (like Zecco was, before Ally bought them).
There's no point in contacting CERT about a Bank of America vulnerability. CERT won't prioritize the report and won't know the right person to talk to, but also, you're a Google search away from finding out who to report to at Bank of America (spoiler: it's Hacker One). These kinds of things don't happen at BofA, not because vulns are hair-on-fire there, but because there's a process in place to handle them.
There's not much point in contacting CERT about a Zecco vulnerability. CERT doesn't know who to contact and doesn't know how to find them and won't spend the time trying. CERT isn't going to publish an unconfirmed report. All CERT is going to do is go to Mitre; you can do that too, and note the guidelines for what will get you a CVE.
The issue here is just TANSTAAFL. It takes a fuckload of effort to triage and confirm vulnerability reports. There's no magic "this is a real vulnerability" certificate you can get CERT or Mitre --- or really anyone who doesn't spend a lot of money to maintain the capability for their own products. If there was, Hacker One wouldn't make half their money selling triage services. :)