As the other answers state, the E-M10 III is not capable of “clean” HDMI output nor USB tethered liveview.
While I know there’s a youtube video about using digiCamControl and OBS as a webcam solution touting it as a solution for any camera, the problem is that guy’s using a Sony camera. The title lies.
digiCamControl relies on a camera’s ability to do tethered liveview. And they do publish a list of supported cameras, which includes zero Olympus bodies. So, even if the E-M10 III did do USB tethered liveview, digiCamControl wouldn’t be your solution.
OBS is a streaming studio application that can take any window (or window area) on your computer and, if you install the correct OBS virtual camera plugin for your operating system, can turn that into a webcam source.
So, if you can find an application for your computer that displays a clean liveview from your camera’s sensor, then you can use the OBS solution. But AFAIK, Olympus does not supply any such application for its users (the ones they do have are either for four-thirds cameras, or (mostly) the latest E-M1 series).
And, given that your camera doesn’t do USB tethering, then possibly your only solution would be if there were some way to send that liveview to your computer over the camera’s wi-fi: something similar to the Olympus Image Share app, but on your computer instead of your phone and that doesn’t draw control overlays over the image. Good luck finding it.