This is possible via an addon.
Exe’s can be launched with nsIProcess. This has the side affect that Firefox semi-waits for response, and i fit doesnt, on osx and linux, their is pid pollution. Meaning the pid of Firefox you launched from, is now on all resources used by the launched exe etc. To avoid this, there is js-ctypes solutions of using ShellExecuteEx on Windows, popen open on OSX, and GIO.launchFileWithURI on GTK/linuxes.
Here’s some detail on stackoverflow: http://stackoverflow.com/a/30468239/1828637
There should be an addon out there that allows this, if not i have plans for an addon that includes such a feature but it’s a little more complex so can’t have it done right away. So if you want i can hook you up with a custom thing, preferably I’d like to show you how to make it.