You could try the User Agent Switcher add-on. The user agent tells the site what browser you are using, with this add-on you can make it appear that you are using IE. There is no guarantee that this will work, if the banks web site uses code that only works in IE, or has ActiveX controls then it will not work in Firefox.

You can try this add-on. You can open sites designed for IE in a special tab in Firefox. I use it to successfully access sites designed for IE, even the Microsoft Update site. The add-on uses IE to load that tab, so be sure that you have the latest, IE8, and all the updates for IE8 from Microsoft.

IE Tab Plus: https://addons.mozilla.org/en-US/firefox/addon/10909/

Link to IE8 download: https://www.microsoft.com/windows/internet-explorer/default.aspx