Do you have anything besides your opinion that can refute and/or dispute the study?
There are plenty of guides and methods for assessing studies as to their likely validity.
This study has several huge problems:
1. The data is self-reported with no standardisation. This type of data collection is known to be highly unreliable - the technical term is "biased". (In the KIGGS study used as a control, the questionnaires were standardised by interviews, where answers were explained, so that a researcher double checked that the correct box had been checked, etc. and that the question or answer hadn't been misunderstood)
2. There is no attempt to check that their study group can be compared to the reference group. Normally, you would check that the sex mix, age, socioeconomic class, schooling, health history (prematurity, etc.) and more were all equally represented in test group and control. Without this, you can't tell if any differences you are seeing come from what you are manipulating "in this case, self-reported vaccination" or from some confounding factor. This is a huge problem in this study, as they openly admit that their age distribution is not even remotely comparable to the reference group. The response of the immune system changes as it grows, matures and becomes more experienced, so that young children (under about 4) have a very different immune response to older children and adults (that's why children get totally different infections to adults). That in itself, should be a big red flag for a study that purports to measure immune activity in children.
3. The population under study is a self-selected sub-group. This, also, is cause for concern as this method of selection often brings with it obscure confounding factors and strong bias.
There are a number of other problems, but those are the scientific ones that just stand out to me.
