To fight the pandemic at home, the United States gave its own vaccine manufacturers priority access to American-made materials needed to make the shots.

As a result, the US government laid claim not only to vast quantities of finished Covid-19 vaccines but also to vaccine components and equipment all along the supply chain, according to a Reuters review of more than a dozen contracts involving some major suppliers.

That has left some countries desperately in need of those supplies to scramble for substitutes, exacerbating international disparities in vaccine access, according to interviews with suppliers, foreign manufacturers and...