The stylelint linter supports linting style parts of Vue single file components. You can run linter on particular vue file as below,
stylelint MyComponent.vue
Other option is configuring stylelint-webpack-plugin in webpack. It can be configured as a dev dependency.
// webpack.config.js
const StyleLintPlugin = require('stylelint-webpack-plugin');
module.exports = {
// ... other options
plugins: [
new StyleLintPlugin({
files: ['**/*.{vue,htm,html,css,sss,less,scss,sass}'],
})
]
}