The Discord integration involves an unofficial approach to getting the token, and is not recommended. The approach is similar to how users can potentially automate their logins ("create self-bots"), which are not allowed by the Discord ToS.
The goal is to move to Discord RPC in the near-term, which supports proxied API/websocket requests with OAuth. Discord RPC is in private beta at the moment.
If you have any suggestions to improve this flow, please create an issue.
Obtaining the token
To set up Discord inside VS Code, you need to have your Discord token. To obtain your token, follow the steps given here.
Instead, log in on the discord client, pop open the web inspector, (ctrl/cmd shift I), and type localStorage.token in the console to get your auth token.
Configuring the token
Once you have the token, run the following commands from the VS Code command palette:
Run Chat: Configure Access Token, select "Discord", and then paste your token in the input box. The token will be saved securely in your system keychain.
Next, you will be prompted to choose your primary Discord guild. If you don't want to select now, you can run the Chat: Change Workspace command later.