They are right
only child or infant fares have to be booked by phone. We are going next month and I have me and my 7 year old dd as adults and got the web fare. I later decided to get a ticket for my infant instead of holding him and when I called to ask about the best way to do it, a nice lady helped me book the infant fare for him because the web fare was sold out at that point.
Actually the outbound flight had the web fare available, but the return flight didn't, so the lady told me to book him on the webfare for the outbound flight and she would simultaneously book the infant fare for the return flight. Saved me $40 which I'm using to get the Scenera car seat for the plane.