I'm trying to have a script that looks like this:

{
    "scripts":
        "setup": "mkdir -p ./my-dir"
}

And it fails, at least on Windows, even if I run it from a Git Bash prompt. Even trying just mkdir ./my-dir doesn't work. I can't figure out any reason it should fail.

The error it gives is a "syntax incorrect" error:

> my-app@0.0.1 stage C:\my-app
> mkdir ./my-dir

The syntax of the command is incorrect.

npm ERR! Windows_NT 6.1.7601
npm ERR! argv "C:\\Program Files\\nodejs\\node.exe" "C:\\Program Files\\nodejs\\node_modules\\npm\\bin\\npm-cli.js" "run" "stage"
npm ERR! node v6.4.0
npm ERR! npm  v3.10.3
npm ERR! code ELIFECYCLE
npm ERR! my-app@0.0.1 stage: `mkdir ./my-dir`
npm ERR! Exit status 1
npm ERR!
npm ERR! Failed at the my-app@0.0.1 stage script 'mkdir ./my-dir'.
npm ERR! Make sure you have the latest version of node.js and npm installed.
npm ERR! If you do, this is most likely a problem with the my-app package,
npm ERR! not with npm itself.
npm ERR! Tell the author that this fails on your system:
npm ERR!     mkdir ./my-dir
npm ERR! You can get information on how to open an issue for this project with:
npm ERR!     npm bugs my-app
npm ERR! Or if that isn't available, you can get their info via:
npm ERR!     npm owner ls my-app
npm ERR! There is likely additional logging output above.

npm ERR! Please include the following file with any support request:
npm ERR!     C:\my-app\npm-debug.log

If this won't work, is there a cross-platform friendly way to initialize an empty directory?

You can use npx plus mkdirp for a relatively lightweight, portable solution without global installs

Add this to package.json:

"scripts": {
  "create-dir": "npx mkdirp _site/assets",
}

And then use like this:

$ npm run create-dir