No no no, it doesn't work that way. These banking websites don't expose a well-defined API that you can emulate. Instead they force you to install a bunch of ActiveX plugins (usually with administrative privilege) and you have to just trust that they won't, say, read the whole content of your hard disk and stream it to a third-party site.
A few years ago (when I was still in Korea), it was usually impossible to open two different bank's pages at the same time: I'd assume that's still the case now. As far as I know, the reason is that both banks will force you to install "anti-hacking" plugins, which hooks directly into your Windows Kernel and makes sure nobody else snoops on what you type. Yes, these webpages try to establish a direct connection between your keyboard and the website, completely bypassing every layer. Now imagine the fun when two such plugins try to run at the same time.
And of course without these plugins you can't use the site. Hell, sometimes these sites spontaneously break just because you're accessing it from the US, because nobody had thought to test them from a client with ping time > 200ms.
Now try emulating that in client. (I don't know if I should laugh or weep.)
EDIT: Besides, if you seriously try to make a platform that can emulate these "security" plugins, sooner or later you will be arrested for making tools to circumvent security measures, and people will be reminded that they should never install anything from "suspicious" sites. (But of course install everything from banking sites.) As a bonus, some news media will claim you were paid by North Korea, and many will believe that.