Actually, I believe you can, and you may win but they still aren’t held liable, the tax payers foot the bill, not the manufacturer. The 1986 vaccine act eliminated financial liability for companies. They have nothing but reputation to lose.
https://www.congress.gov/bill/99th-congress/house-bill/5546
And from Wiki,
The National Childhood Vaccine Injury Act (NCVIA) of 1986 (42 U.S.C. §§ 300aa-1 to 300aa-34) was signed into law by United States President Ronald Reagan as part of a larger health bill on November 14, 1986. NCVIA's purpose was to eliminate the potential financial liability of vaccine manufacturers due to vaccine injury claims[1] to ensure a stable market supply of vaccines, and to provide cost-effective arbitration for vaccine injury claims.[2] Under the NCVIA, the National Vaccine Injury Compensation Program (NVICP) was created to provide a federal no-fault system for compensating vaccine-related injuries or death by establishing a claim procedure involving the United States Court of Federal Claims and special masters.