Slack doesn't provide an easy way to extract custom emoji from a team. (Especially teams with thousands of custom emoji) This Gist walks you through a relatively simple approach to get your emoji out.
If you're an admin of your own team, you can get the list of emoji directly using this API: https://api.slack.com/methods/emoji.list. Once you have it, skip to Step 3
HOWEVER! This gist is intended for people who don't have admin access, nor access tokens for using that list.
Follow along...