I've been to 'Ohana a few times with parties as big as 20 or 30, so they can accomodate your group with a single reservation.
However, you should call ASAP, because you're already well within the 180-day mark, and 'Ohana fills up fast, especially in June.
One other thing to keep in mind is that for large groups the Disney Dining Line will sometimes tell you to call the restaurant directly and make a ressie with the manager, so you might have to do it that way; call the Dining Line first, however.
If you do have to call the restaurant directly, don't be surprised if no one answers; just leave a message stating, "Hello, I want to make a dinner reservation for a group of 16 people on June XX. My name is ______ and I can be reached at xxx-xxx-xxxx." Oh, and don't speak too fast when giving your info (pet peeve of mine - I hate having to replay a long phone message 4 times to get a number or name!) A manager will usually get back to you later that day; I've had them call me back 10 minutes after I left my message!
Be prepared not to get your #1 preferred date and time, and be ready to try different dates and times. With a large party, flexibility is key, especially once you're so far inside the 180-day booking window.