As a summary, the various vaccine manufacturers got the federal government to agree not to ship them outside the US as a condition of purchase, because then there would be additional liability issues the manufacturers would be exposed to.

This seems like an easy fix. Coordinate with those other country's governments pass a similar indemnification law for the COVID-19 vaccine and the price for the US organizing that and the other governments doing that will be the manufacturers allowing the US to ship its doses to them.