这些网站最有可能会抓住个人资料页面，这是不允许的。不要这样做，反正应该不用了。只需让用户授权您的应用程序，您将获得（应用程序范围）的ID。没有严格的使用用户名或其全球ID的申请。

I would like to retrieve a Facebook user's id given their username.

Example: Given helloworld return 100003263065603.

I understand that the Graph API does not allow this anymore, but several sites, such as lookup-id and findmyfbid are able to do this.

I am guessing they are doing some sort of scraping? If so, how would I implement this in Node.js

Those sites most likely scrape the profile page, which is not allowed. Don´t do that, should not need to anyway. Just let the user authorize your App and you will get his (App Scoped) ID. There´s no serious application for using the username or his global ID anymore.