I don't think that unvaxed children never get sick.... I just think they have a better chance of resisting whatever is circulating around, and tend to recover faster when they do become ill.
My DD is 2.5yo, received pumped breastmilk for three weeks due to her complete and utter refusal of the breast despite all attempts, and formula fed thereafter. I started running a home daycare when she was 6mo up until last month with up to five extra kids here during the week. We attend a number of activities and she goes to Sunday School, so is around at least 30+ different kids each week.
She has had a few mild colds. Maybe two each year? I have never counted but I was never concerned about the frequency. Each lasts only a couple of days and she is back to her old self. She has never had the lingering cough of constant green snot that I have seen in countless other children.
We had one puking experience that lasted about three hours, and again back to her normal self.
The worst thing she has had was last summer when she had Hand Foot and Mouth. That came with a very sudden fever of 104F and she felt pretty crumby, and then we had a couple of sleepless nights when the sores on her feet were really itchy. And then she was fine. (I, on the other hand, came down with it a couple of days later and was laid up for a week.)
I would say that all of the kids I know who seem to be constantly sick are fully vaxed. One little guy in my daycare was sick with, and then passed on to me, six separate colds, two stomach flus and the aforementioned Hand Foot and Mouth all between August and December. He also had a couple of ear infections during that time. I picked up nearly everything he got because he was always coughing and sneezing in my face, but DD missed most of it. I wish I had her immune system. I closed my daycare just before Christmas and this has been the longest I have been healthy in months.