A solution without additional extensions

With sass

Assuming you have sass installed globally with for instance:

npm install -g sass

Open the folder and create a task.json file under .vscode containing

{
// See https://go.microsoft.com/fwlink/?LinkId=733558
// for the documentation about the tasks.json format
"version": "2.0.0",
"tasks": [
    {
        "label": "Watch Sass",
        "type": "shell",
        "command": "sass --watch src/style.sass styles/style.css --style=compressed",
        "problemMatcher": [],
        "group": {
            "kind": "build",
            "isDefault": true
        },
        "runOptions": {
            "runOn": "folderOpen"
        }
    }]
}

With node-sass

Replace sass with node-sass in the above.

In both cases make sure the source/destination filename, location and extension are correct (in my case src/style.scss and style/style.css)

With a Workspace file

Or copy the section in your .vscode-workspace file to avoid clutter of .json files.

Make sure to change the sass source and destination files to your personal needs.

Setup VSCode

[EDIT] whith the current version this is asked the first time you open the workspace file and the following steps are no longer needed. To a llow automatic run tasks

1. Ctrl+Shift+P
2. select Manage automatic Tasks and
3. select Allow Automatic Tasks in Folder and
4. close and reopen your folder (or Workspace)

The sass compiler will be called and starts watching all your edits with a reassuring:

Compiled css\src\style.sass to css\style.css.
Sass is watching for changes. Press Ctrl-C to stop.

or with error messages when compilation failed.:

Error: semicolons aren't allowed in the indented syntax.
  ╷
7 │     padding: 0;
  │               ^
  ╵
  css\src\_base.sass 7:12  @import
  css\src\style.sass 1:9   root stylesheet

EDIT command and args can be separated

{
    "label": "Compile sass",
    "type": "shell",
    "command": "sass",
    "args": [
        "--watch",
        "--style=compressed",
        "./style/src/main.sass",
        "./style/main.css"
    ],
    "runOptions": {
        "runOn": "folderOpen"
    }
},

Source - https://pineco.de/the-simplest-sass-compile-setup
By - Adam Laki

Make sure your project has a package.json file (and you have Node installed on your machine). Run npm init if you have Node but not package.json file, to initialize one.

Install sass package:

npm install sass --save-dev

Learn more about the package and its CLI

In the package.json file's scripts section, add these:

"scripts": {
    "sass-dev": "sass --watch --update --style=expanded assets/css",
    "sass-prod": "sass --no-source-map --style=compressed assets/css"
},

Run scripts:

npm run sass-dev
// or
npm run sass-prod

What is a source map? A particular file that allows the browser to map back from the processed, concatenated files to the original ones. It is helpful because we can see the original file names when we debug the CSS in the developer tools.

The above is a very basic setup to compiling SCSS files and "watching" them for changes. Your server should have a pipeline or some sort of build system that it would be able to run this npm command in order to compile SCSS files, so theoretically you don't need to push your pre-compiled CSS files to the server, but it does it by itself.