This is what I say in arguments about God. The atheist will say such things like "Why would a loving God give a child cancer?"
"God doesn't give children cancer, but vaccines do!"
SIDS mortality began rising before the modern childhood vaccination schedule expanded substantially. In an analysis of U.S. data from 1968–2009, SIDS mortality increased from 1968–1979, then began declining from 1979 onward.
The principal DTP series was already routinely given at 2–6 months by the early 1980s—the same age range in which SIDS naturally peaks. CDC explicitly noted this overlap in its 1984 guidance and cited case-control evidence finding no causal relationship between DTP and SIDS.
The really dramatic U.S. SIDS decline occurred around 1992–2000, coinciding with the recommendation to put infants on their backs to sleep in 1992 and the nationwide Back-to-Sleep campaign in 1994. CDC says the SIDS rate subsequently stabilized in the 2000s even as childhood immunization increased.