Yes, you have to call. I found this out when trying to book 4 award tickets (using 3 different accounts) for hubby and myself and our 2 minor sons.
The airlines require you to call. If there is a fee for booking by calling (which many do) ask them to waive the fee because you HAVE to call to book for the minor (and the adults travelling with in our case). It is standard for them to waive the call-in booking fee in this case, but if you don't ask them to waive it they might not????
Anytime you have tickets (award or paid) where you have your kids flying on a separate reservation than yourself (or even different family members), call the airline and have them "link" the reservations. This way if something happens (flight is cancelled and you are rebooked, seats get switched, flights change, etc) they will keep those reservations together. I learned this when my 2 boys (very young at the time) were on a separate reservation than hubby and myself, and the flight was changed so people on the flight were put on 2 different flights. Kids on one and us on the other. I caught the change and called and had them linked and no problems after that. Also learned this the hard way when our flight out of Munich, Germany was cancelled (plane problems) and everyone was rebooked on different flights. My elderly father was put on a flight connecting through Amsterdam and arriving in Detroit 1 hour before us. I hadn't thought to link our reservation to his, so we were put on a flight going through Paris. We VERY NEARLY missed our connection in Paris....and we had the car keys! If my husband hadn't thrown a fit in the Paris airport, we wouldn't have arrived home until a day later....I can't even imagine what my dad would have done!