I click on the .exe file, and immediately get a popup 'Webcam Wallpaper has stopped working' which immediately closes.
You receive errors because Webcam Wallpaper isn't supported anymore. I tried to download the installation package available from the application page at Software and although it installs properly, there is an error when you try to apply it. It says that it connects to the update server, but the connection fails because the software isn't updated anymore.
That error could mean that you lack some components as well. For example, install Visual Runtime C++ and NET Framework.