If you can't find a Chicago death certificate from 1878-1922 for free at FamilySearch's Record Search it may be that the individual had a coroner's death certificate. An easy way to check is to search the Coroner's Inquest Index. (see link below)
If you find the name there, I can often locate the corresponding death certificate (different from the inquest record you'd get from IRAD at NEIU). Turnaround time is usually a few days.
Please note: Death certificate forms before 1910 did not ask for parent or informant names and forms before 1916 did not ask for spouse.
Fee if I find the record you requested: $8.00
Fee if I don’t find the record after a thorough search: $2.00
Please provide the full index entry from the Illinois Statewide Death Index or tell me with the name and death or burial date (at least a year). If you need more than one record, please create a request for each one separately.