I had to figure this out the other day. Here is how it worked for me:

1. Make sure you have Japanese - Microsoft IME enabled in your language bar.
2. When you have FFXIV open, switch the language input to Japanese (if you are in windowed mode, it's easy to do, go click the EN and switch it to JP)
3. You can only type Japanese AFTER you have hit enter in your chat box to start chatting. IE. You can't type in Japanese from the getgo, you need to press ENTER or the / key to open the chat box. You will need to press ALT+~ to switch to Hiragana, and ALT+SHIFT to get back to Romaji input
4. Switch back and forth as needed!

I hope this helps.