What countries have sovereign immunity in the US?

None, only diplomats have limited immunity from prosecution in the USA. This is referred to as diplomatic immunity rather than sovereign immunity.

Sovereign immunity in a feature of Common Law in which the sovereign or the state is immune from civil or criminal prosecution. The answer to your question is then that the American Federal government possess sovereign immunity except in those cases in which that immunity has been waived by statute. While not countries in themselves the constituent states of the United States are presumed to possess sovereign immunity.

A related concept is that of extraterritoriality or exemption from the jurisdiction of local law. This may be negotiated on the occasion of the visit of a head of state or as part of a status of forces agreement covering the armed forces of one nation stationed in another nations territory.