Which key is the 'Take Screenshot' command actually assigned to?

That's under Config / Misc. 2 / Commands.

If it's something like [ and a keyboard layout in use is one where you need a modifier key such as alt_gr to make it happen, it might not work, and changing it to something else (such as F11 perhaps) should help.

While on the topic, you might like to change the screenshot directory to something else if it still defaults to the install path, and in addition to that, enable 'Take a screenshot in your screen's resolution'. Both of these need to be done via the FFXI Config utility.