There have been issues more recently with vaccine safety. Or at least, there were enough lawsuits to start frightening off vaccine manufacturers. Since 1986, however, they've been insulated from lawsuits under federal law:[0]
> The National Childhood Vaccine Injury Act of 1986, ..., as amended, created the National Vaccine Injury Compensation Program (VICP), a no-fault alternative to the traditional tort system. It provides compensation to people found to be injured by certain vaccines. Even in cases in which such a finding is not made, petitioners may receive compensation through a settlement.
> The VICP was established after lawsuits against vaccine manufacturers and healthcare providers threatened to cause vaccine shortages and reduce vaccination rates. The Program began accepting petitions (also called claims) in 1988.
The current anti-vaccination movement began largely over concern that thiomersal preservative in many childhood vaccines was causing autism. Although multiple epidemiological studies indicated that there was no appreciable risk, "the Centers for Disease Control and Prevention (CDC) and the American Academy of Pediatrics (AAP) asked vaccine makers to remove thiomersal from vaccines as quickly as possible as a precautionary measure", and it was ~gone by 2001.[1]
0) https://www.hrsa.gov/vaccine-compensation/about/index.html
1) https://en.wikipedia.org/wiki/Thiomersal_and_vaccines