After a holiday break and other things, Ive finally turned my attention to this and I believe I have something that works. @LeventK is correct in saying that Google Apps Script is the way to go. You need to call the web service using the google function UrlFetchApp, passing it the url and options, which itself can include authorization strings and an optional payload (JSON-format data).
Inside the script, you should also be able to call AppSheet API to update spreadsheet (Google sheets). I dont think you can update a rdbms datastore via the appsheet api.
It works, and there should be examples, etc. on the web, but if you get stuck I’ll see what I can do in the way of example code