"The majority of parents reported believing that vaccines were important to children's health (79.8%) and that they were either confident or very confident in vaccine safety (79.0%)." So at least 21% of parents are not confident in vaccines or find them to be safe. That is not a small number.
Here are some more polls and surveys.
CU med study: Parents grow more wary of vaccination safety
Parents' Vaccine Safety concerns dominate Office Visits
A new study published online April 15 in the American Journal of Preventive Medicine shows that 53% of family practice physicians and pediatricians surveyed in 2009 report spending 10 to 19 minutes on vaccine discussions when parents harbor "substantial concerns" about adverse events. For another 8%, these conversations last 20 minutes or longer.
Harris Poll Shows Majority Support Parental Vaccination Choice, Say Pharma Has Too Much Influence
Associated Press, "1 in 4 parents think shots cause autism."
T.Neale, "Vaccine and Drug Safety Top Parents' Research Priorities," MedPage Today, October 12, 2010.
Public wants inquiry into MMR
Scientific Fraud May Be More Widespread Than Thought, Poll Says