In 2016 Windows launched the Anniversary Edition Windows Update. In this update, Windows removed support for the compression streams used by a majority of Webcams. For the majority of users this update installed automatically, which then led to the error shown below:
Fortunately Windows released a patch for this update in September 2016 which resolved this issue. This is update KB3194496. The easiest way to install this update is to run Windows Updates continuously until there are none left to install. At this point the Logitech App and other programs using the Webcam should work as they did before.