Creating an organization is web-only on GitHub.com (from GitHub 29 's documentation; not run here): profile picture, Settings, Organizations, New organization; pick a plan (Free is enough), enter a free name, a contact email and whether it belongs to you or to a business (which accepts the Corporate Terms of Service), then invite members and click Complete setup. The only create endpoint, POST /admin/organizations, exists on GitHub Enterprise 29 Server alone. With gh 29 (Installing gh) you can see both facts:
gh api user/memberships/orgs
echo
gh api --silent -X POST admin/organizations -f login=booknest -f admin=binarybehemoth[] gh: Not Found (HTTP 404)
Everything inside an organization is scriptable once it exists. On day one, require two-factor authentication for all members, set the base repository permission to Read or No permission, verify your domain, and add a second owner so that one lost account can never lock the organization out.