As far as I know you can’t
If the download linked at https://developer-rig.twitchcdn.net/Twitch%20Developer%20Rig%20Setup.exe from https://dev.twitch.tv/docs/extensions/rig/#extensions-developer-rig doesn’t work then you are out of luck and really should upgrade your OS.
Windows 7 left mainstream support January 2015
And leaves Extended support January 2020
You should upgrade from Windows 7.