Unless they have relationships with the banks now, and would be given a heads-up. Perhaps that's the case. It sounds like it may be.
Yes you can, and banks already do this. Even the E-Trade app refuses to work unless it's on the most current version. When you control the apps and the API, you can pretty much do as you please.
In other words, many of the big banks have such a dysfunctional relationship to internet banking that customers of many of them have learned to deal with not being able to use the apps for long periods of time - it's hard to imagine that a third party will be perceptibly worse.
Not to mention, once they discover what changed, now this service has to go make the changes on their system... meanwhile that bank doesn't work with this service anymore.
So the App is broken for some undefined period of time.
"All" of their client code is a bunch of apps, that mostly auto-update. My bank (small town credit union) has a banking app that refuses to work at all unless it's the most recent version of the App. I see no reason why this would be a challenge for the bank - it's only a challenge for this service.