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.