With the current measles outbreak this is a “live” issue. From Dorit Reiss at UC Hastings College of the Law:
Further, we do make exceptions and impose liability for nonaction when there are strong policy reasons. When parents decide not to vaccinate (absent a valid medical reason), they are choosing a bigger risk for their own child and rejecting expert opinion. That is bad enough; in a real sense, they are failing their own child. But they also put others at risk, others who did not choose that risk. Under these circumstances, there are good reasons to create an exception and find that parents who do not vaccinate violate a duty of care and should take personal responsibility when anyone is harmed. This social choice can be made easier by legislative action. State legislatures can create laws that impose liability when non-vaccinating is shown to cause harm.
From a 2014 piece by Blake Simpson, then a J.D. Candidate at the University of Nebraska College of Law:
In response to the problem of decreasing vaccination rates in Ashland and other communities with high non-medical exemption rates, bioethicist Art Caplan has advocated for using tort law as a policy-shaping tool to help achieve public health goals regarding vaccine mandate compliance. Caplan believes that statutory religious and philosophical exemptions should still be available to parents, but that liability for negligence should flow from any harm caused by their decision to withhold vaccination from a child regardless of whether an exemption has been procured. As Caplan explains,
If your kid gets the measles, and remember public health officials are getting very, very good at tracing outbreaks to their source, and makes my kid sick (can happen since vaccine is not 100% effective), my newborn baby die (newborns can’t benefit from vaccines) or my wife miscarry (fetuses are at especially high risk), then shouldn’t I be able to sue you for the harm you have done?
In a typical case, the plaintiff would shoulder the burden of proving each element of a traditional negligence claim in order to recover against the parent of an unvaccinated child: duty, breach of duty, causation, and damages. As to duty, Caplan contends that parents have a general duty to prevent foreseeable harm to others. Because the failure to vaccine could result in reasonably foreseeable harm to the child and others, this duty applies regardless of whether a parent has obtained a religious or philosophical exemption. Caplan believes that exemptions do “not negate the fundamental duty one has to act reasonably in preventing the spread of disease to others,” and the failure to vaccinate a child represents a breach of duty regardless.
And from a University of Cincinnati Law Review, Vol. 82  article by Teri Dobbins Baxter:
Parents have the right under current state and federal law to choose not to immunize their children. Their choice to exercise this right should not expose them to tort liability. However, their choice, and the constitutional and privacy rights implicated by the choice, do not absolve them of their duty to exercise ordinary care to prevent causing harm to others. Allowing those who have been infected by unvaccinated children to pursue tort claims merely recognizes this duty. While courts have not addressed tort claims or duties in this precise context, holdings in other cases involving negligent transmission of contagious diseases support the conclusion that public policy favors tort liability. It promotes the compelling state interest in preventing the spread of disease without unduly infringing on the right of parents to direct the care and upbringing of their children. For these reasons, tort liability should be available against parents who choose not to immunize their children and who fail to use due care to prevent those children from contracting harmful diseases and infecting others.