There is no evidence for this. The claim is that immunisations and SIDS rates both increased in USA so must be linked. But the rate of SIDS has been decreasing for some time, while the rate of immunisation has been increasing.
Babies receive several vaccinations between two and four months old. This happens to be the same age that they are at the greatest risk of SIDS. Multiple studies have been carried out to examine the relationship between the two. Each one has come to the same conclusion, that vaccines do not cause SIDS. There is no causal link between the two.
The most significant study was carried out in 2003 by the National Academy of Medicine (then known as the Institute of Medicine). It found no increased risk of SIDS after immunization. More than 129,000 children were examined as part of the study. In fact, general observations appeared to be that vaccines reduced the risk of SIDS.
This is also supported by the numbers. In 1999, around 5,000 infants in the US died from SIDS every year. A decade later, in 20099, that figure had fallen to 1,900 a year, despite vaccine use increasing.
A far greater contributor to SIDS is the baby’s sleeping position, whether there is a smoker in the house and bed sharing. Putting a baby to sleep on their back significantly reduces the risk of SIDS.