Death records for Lorain County are kept by various Health Departments. If he died in the city of Elyria or Lorain, you need to contact the Health Dept. of the appropriate city. If he died anywhere else in the county, the death record is kept by the Lorain County Health Dept.
Go to the Lorain County GenWeb site listed below. The addresses of the various Health Departments and specific ordering information, including the cost of the copy of the death certificate is given under the heading "Resources Available for Lorain County".