The Great Chicago Fire of 1871 destroyed a majority of documents and registers, so no vital records exist prior to then. Both birth registers (certificates exist post-1878) and marriage licenses began again in 1871, whereas the death certificates didn't begin again until 1878.
|Image source: Hearst's Chicago American, January 2, 1902, Page 8|
(Click to enlarge)
How To Order Copies:
Birth certificates and registers are available on microfilm from 1871-1922 and can be ordered through the link here.
Death certificates are available on microfilm from 1878-1945 and can be ordered through the link here.
Marriage licenses are available on microfilm through 1871-1920 and can be ordered through the link here.