I wasn't thinking about the LucidChart per say. I was wondering how an Web based-app could be intergrated into Excel such that a user was prompted to login and then import user specific data directly into a Excel spreadsheet range using an REST API.
Nowadays XHR is a bit limited as far as exception handling and other niceties you'd like for API calling. But Excel VBA can access anything through COM interop, such as a more robust C# library using .NET objects like HttpWebRequest.
Excel also has a notion of web queries, to populate a range by making an HTTP request and scraping a table out of the HTML. I think that existed as long ago as Excel 2000 and was brought out to the UI in 2007. It doesn't play well with websites requiring a login or a nontrivial click path, though. I have vague memories of wrestling with this, through the spectacular method of launching an IE window from Excel and relying on the MSHTML layer to share the login cookies between that and Excel!
I can't speak to Excel integration, but I think people are doing that.