There are some suites you can book online, however I think it depends on the type you want. For example at the Contemporary I have seen Garden Suites and One Bedroom Suites on the Tower Club Level available to book via the WDW website. So my guess is if it's not listed on the WDW website no matter what days you put in, call CRO and make the reservation threw them.
The King Kamehameha suite at the Polynesian which is the best you can get I have not seen online to make a reservation. So if you or someone else wanted to make a reservation for that one, then you would need to call CRO.