The Lake County Court house established in 1841 burn down in 1875 so if the family did register the death then you need to check with the court house today or a local Historical Society if any ledgers survived. Not just death but probate and wills.
http://lakeco.genwebsite.org

Yes a family up to the time a state required deaths to be registered and the use of established local cemeteries for burials, could bury family members on private land.

Check to see if any local records including church membership were microfilmed by WPA in the 1930s. An LDS FHC should have films to rent, catalog online at www.familysearch.org