If you're using the Game Installer and not the Browser to play the game you need to install this Adobe Flash Player for Firefox from Adobe website.
https://get.adobe.com/flashplayer/otherversions/
Step 1:
Operating System: Select Windows 10
Step2:
Select a version: FP 32 for Firefox - NPAPI
Follow the prompts. Should be all set to run The Settlers Online client.
If your using VLC Player it might conflict with the Settlers client. Here are the steps:
1. Uninstall VLC Player.
2. The Settlers client should work now.
3. If you need VLC Player, re-install it but when you get to the Prompt of selecting options, uncheck the following.
- Mozilla Plugins
- ActiveX Plugins
4. VLC Player and The Settlers Online client should work without conflicting.