This extension is published under the Mozilla Public License, an open source license, and the code repository is at https://github.com/jscher2000/Save-webP-as-extension/. You also can review the code by right-clicking the "Add to Firefox" button and using Save Link As. The .xpi file is a ZIP archive, so you can extract it and read the files locally.

The reason such broad permissions are required is that many images are hosted on a different server, and Firefox doesn't provide a convenient way to prompt for permission to operate on embedded cross-site content. Maybe I can make a lower-permission version that only works with stand-alone images?

If you prefer an extension-free approach, you can emulate the way this extension works using Firefox's built-in screenshot feature. That usually downloads in PNG format, but occasionally it may use JPEG format.

On all sites? I wonder whether you have a setting that conflicts with how this extension works. Generally speaking, it creates an HTML canvas the same size as the image and puts the image on the canvas, then extracts a blob object that it can save in your choice of formats. Similar to screenshotting. If you want to try some troubleshooting, you could use either contact method on the main listing page: (1) email, or (2) create an issue on Github.