Yes, all the scripts built with Laravel + Vue.js supports REST API. Here are some list of the APIs which are usually available in all the scripts:
http://docs.scriptmint.com/laravel-vuejs-starter-kit
Visit this URL and click on the API tab in the sidebar.
Click on the "Show Params" button to know more about the parameters to be passed. Click on the "Show Reponse" button to know more about responses.