Is it possible to use google apps script in my HTML? I want to be able to write to a spreadsheet from a form in purely Javascript from an external framework such as Node.js.
Google Apps Script's syntax is Javascript, however it is a unique server-side framework that does not behave as a library to applications outside of the Apps Script servers. (No, you won't be able to use Google Apps Script in your node.js app.)
However, that doesn't mean that your node.js app (or any other app on the web) can't interact with your spreadsheet. For instance, your app could authenticate as you using the OAuth API, then access the spreadsheet through the Google Drive API. For an example of this, see Accessing Google Spreadsheets from Node.js
Alternatively, you could roll your own spreadsheet API in Google Apps Script, to support read / write of your sheet via HTTP requests from your node.js app. There are plenty of examples of that, for example Insert new rows into Google Spreadsheet via cURL/PHP - HOW?.