The EM system for a medical emergency is explained here. Key roles for state and local government are described. See: If the coronavirus hits America, who’s responsible for protecting you? What the response to a Covid-19 outbreak in the US could look like.
Update: Here is another useful article about local EM involvement. Coronavirus: How Local Officials and Emergency Managers Respond
Emergency managers can and should help by lending their expertise in providing shelter and broadcasting good information to the public. Seattle use its AlertSeattle to inform residents about a local case.