This question arises from a total misreading of legal terminology. From NOLO:
A product that is unavoidably unsafe cannot by its nature be made safe and still fulfill its intended purpose. In other words, if the product were to be made safe it would no longer be capable of being useful for its original purpose. For example, a kitchen knife may be used to cut foodstuffs. It is a sharp object that is capable of causing injury. If it were to be made safe, and the blade made so blunt that it could not cause injury to anyone under any circumstances, it would no longer be able to fulfill the role for which it was intended.
The phrase "unavoidably unsafe" has indeed been applied to vaccines, for example the live polio vaccine. But it should not be necessary to cite case law to prove this, because if you read the description above with a knowledge of how vaccines work, this is obviously a literal description of many vaccines, especially live vaccines.
The question is, if you are injured by a vaccine or another medical treatment, whether this constitutes a tort which entitles you to legal compensation. In the case of an "unavoidably unsafe" treatment, the answer is that the risk of the treatment could not have been avoided by the medical practitioner so you must work harder to demonstrate a tort.
This concept has value for lawyers, but it doesn't make sense to deploy the term "unavoidably unsafe" in deciding whether to accept medical treatment. Would you refuse surgery because scalpels are unavoidably unsafe? This is far from the most important thing to take into account when evaluating the risks of surgery.
Another unavoidably unsafe product is blood:
Victims of transfusion-related diseases, however, generally have been unsuccessful when making claims against the purveyors of blood products because of blood shield statutes that were initially enacted in response to unknown pathogens that made the blood an "unavoidably unsafe" product.
If you were dying from blood loss, would you refuse a transfusion because the "unavoidably unsafe" status of blood makes you unable to sue the hospital if you accidentally contracted AIDS? Most patients would not really place urgency on this legal issue.
The answer to the question is yes, but it is not evidence against vaccines.