I found a guide on another RPG Maker website. The guide explained that you could put in a mp3 and it will play it in the sig, and I tried that.

But, instead, I wanted a video. Is that allowed? What I really wanted was the first Bleach opening to play, but I couldn't find it.

Could I still leave the YouTube video embbed in my sig?

Don't do that.
It takes up unnecessary bandwidth, slows the loading time of the webpage, and can generally just annoy a large population of posters on any given forum.

Yeah, for the reasons Stythe stated, that's pretty much a "Hell No."

However, we do appreciate you asking before doing so.