I'm trying to add a feature to my website as follows:
Clicking a button appends text to a Google document.
Obviously I will need to create an Apps Script in the drive. The question is how to trigger the Apps Script from my website. You can assume that I am the owner of the drive/document and so have permissions to edit it in any way I like.
I have looked at these topics:
It seems they are all actions performed from the drive itself and not triggered remotely.
I have also looked up the Apps Script API in Google but have not found a way to do this. Is it even possible?
Yes, it is possible.
First write an Apps script that changes your desired document. Then deploy it as a web-app running as you that anyone has access, even anonymous. Check out the guides at Apps Script page to see how to write your script as a web-app.
Your script will then have a public url, which you can call from your website and have it run "remotely" normally.