Fear of vaccination is quite wide spread. Vaccines are linked (albeit research does not support this link) to autism, but they were some time ago also linked to SIDS incidences. Are vaccines safe to your child? It turns out that yes, they are, and careful research ruled out immunization as a possible cause of sudden infant death syndrome. Even more, vaccines were thought to lower the risk of SIDS:

Immunisations are associated with a halving of the risk of SIDS. There are biological reasons why this association may be causal, but other factors, such as the healthy vaccinee effect, may be important. Immunisations should be part of the SIDS prevention campaigns. [Vennemann et alo, 2007]

This is related to currently prevailing hypothesis about SIDS causes. Paul Goldwater published w 2011 a review on SIDS (Goldwater, 2011) in which he assessed all the hypotheses of SIDS mechanism according to three questions:

1. Does the hypothesis take into account the key pathological findings in SIDS?
2. Is the hypothesis congruent with the key epidemiological risk factors?
3. Does it link 1 and 2?

Some hypotheses did not meet any the criteria and only one (infection) met all three. That might explain apparent lower ratio of SIDS incidences in vaccinated children – stronger immune system might extert protective effect on an infant. This link is putatively confirmed in the 2015 study by German researchers who studied specifically  diphtheria-tetanus-pertussis immunization and its putative role in SIDS:

Increased DTP immunisation coverage is associated with decreased SIDS mortality. Current recommendations on timely DTP immunisation should be emphasised to prevent not only specific infectious diseases but also potentially SIDS. [Müller-Nordhorn et al, 2015]

Other studies claim that this is not the case, as no correlation between vaccination and sudden infant death was observed after correcting for confounding variables:

There is no increased or reduced risk of sudden infant death during the period after the vaccination. The previously reported protective effect seen in case contol studies is based on the inclusion of unvaccinated cases.  [Kuhnert et al, 2012]

Whatever hypothesis about protective (or lack of protective) effect is true, it is absolutely clear that vaccines do not cause SIDS. Having your child immunized will protect it from other diseases, but will not have a negative effect when sudden death is concerned.