You don't need to replicate all of the functionality that activex has. You just need to emulate the behaviour from the banks point of view of whatever their particular activex does. Maybe the ActiveX rewrites a bunch of files with admin privileges "for security", and then negotiates some sort of key exchange... in that case just write JS that says it did the shit that requires admin privileges, then negotiates the key exchange).

So the question is, if you are willing to ignore their activex and run your own custom JS instead, could these websites be made to work?

This is completely false. Firefox extensions have access to everything the browser does and can contain arbitrary native code.

There's a reason Mozilla's addon repository has a review process.

What makes addons better than ActiveX controls is that they're understood to extend the browser for the user's benefit, not merely make a crappy website work. Fewer people would buy that a bank website needs to install a Firefox extension before you can log in.

Oh $BABY_DEITY no! That's not what I was suggesting! Activex in FF? Scared the hell out of me...

What I meant is either reverse-engineering the algorithm and reimplementing it in FF chrome in clean JS like jlgreco mentioned below, or including just the kernel of the important logic, the way ndiswrapper uses windows driver for communicating to the network cards without actually implementing whole windows kernel.