[Vuejs]-Vue webpack build – Failed. You may need an appropriate loader to handle this file type

It looks like you need a different style loader for vue templates.

Taken from their documentation

Note: You might need to install that dependency vue-style-loader

In your webpack config file.

      {
        test: /\.css$/,
        use: [
          'vue-style-loader',
          'css-loader'
        ]
      }