The Civ III World Editor is included in the Steam version of Civ III Complete. However, it cannot be accessed through the Steam interface. It must be located within the games directory and launched from the explorer.

Navigate to:
C:\Program Files (x86)\Steam\steamapps\common\sid meier's civilization iii complete\Conquests

Launch the Editor by executing Civ3ConquestsEdit.exe

“Alternatively, you can navigate to actual scenarios or conquests produced by somebody by double clicking the Conquests or Scenarios folders and then navigating to the scenario or conquest you wish to see through the editor” (Darb05, 2012).

Thanks so much to Darb05 for posting this information in the discussion forums. This guide is intended to preserve the information he provided and I would like to acknowledge this user as the source of this awesome bit of info.