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VS Code's builtin language features can format HTML, JS, TS, and JSON. You can use these settings:

"[html]": {
    "editor.defaultFormatter": "vscode.html-language-features"
},
"[javascript, javascriptreact, typescript]": {
    "editor.defaultFormatter": "vscode.typescript-language-features"
},
"[json, jsonc]": {
    "editor.defaultFormatter": "vscode.json-language-features"
}

I saw in the HTML docs that VS Code uses js-beautify under the hood. js-beautify can also format CSS. However, if I open a CSS file, and run the the command "Format document with", it does not offer a builtin formatter. Am I right in thinking that there is no builtin CSS formatter then?

There is no mention of this in the CSS docs.

It seems a bit strange considering that for HTML files, the CSS in style tags is formatted.

Edit - 1/6/22

In v1.66 (March 2022), the built-in CSS extension now is a formatter. The formatter works with CSS, LESS, and SCSS. It is implemented using the JS Beautify library. 🌟

Answers

See GH issue: [css] provide css formatter one is probably coming to v1.66 (it is in Insiders now).

and enable:

CSS > Format: Enable enable/disable the default css formatter, enabled is the default option

Other options:

CSS > Format: Newline Between Rules Separate rulesets by a blank line.
CSS > Format: Newline Between Selectors Separate selectors with a new line.
CSS > Format: Space Around Selector Separator Ensure a space character around selector separators '>', '+', '~' (e.g., a > b).