[Vuejs]-Setting up a custom linter in WebStorm (IntelliJ)

Your settings are wrong. You have to specify the ESLint package (like path/to/your/project/node_modules/eslint), not its plugin as EsLint Package: value in Settings | Languages & Frameworks | JavaScript | Code Quality Tools | ESLint. It’s ESLint that loads the plugin and lints your files, not the plugin itself