I suspect it will be pretty difficult to log in this specific website from a Delphi application.
You will need to log via a Google account, and handle cookies as expected.
Therefore, I'm afraid you won't be able to use any direct Indy / Synapse / ICS / WinINet library to access the data.
You will probably need to embed a full browser within the application, then access to the content from the JavaScript point of view, remoting the browser from your application.
Take a look at those components:
TWebBrowser
(i.e. Internet Explorer) as available in Delphi;
- Delphi Chromium.
Then you will have to parse the JSON content. But you must first ensure that you can login the application!