There doesn't currently seem to be a way to access the spritesheet of an animated emoji once it's been created/uploaded.
I find it odd that once you've taken photos for an animated emoji in-game and it's been turned into a spritesheet there's no way to properly recover the spritesheet, whether it be for safekeeping or editing the timing of the frames.
I would really appreciate if there were a way (perhaps on the website) to access the spritesheet from an animated emoji.