A new experimental homebrew is currently in development, which allows users to play videos on their Nintendo Switch. It's called PlayerNX, and though progress is still early on, you can play .MP4 files if you add them to a sdmc:/media/video.mp4 directory. It includes basic functions like fast-forwarding, rewinding, and pausing. There's no support for audio at the time of initial release, but this project does show that a video player can be made, and is a notable step forward.
You can download version 0.1 on GitHub below.