I would like to be able to use an .mp4 and/or .webp as the cover image for posts. The limitations to the file types that can be uploaded is preventing me from doing so.
Is there a way to change the allowable file types in Self-hosted?
Is there a specific file in which that whitelist exists?
Any hints as to how I may proceed would be very helpful. 
.webp is already in the allow list.
mp4 is a video/audio format, and Ghost is not designed to display a video or audio file as the feature image - it needs to be an image. Even if you edited the allowed filelist here you wouldn’t get the behaviour you’re looking for.
There are quite a few threads on the forum with solutions for using a video instead of a feature image: