gistfile1.txt

// JS File -- index.js

Vue.component( 'test-com', {

    template: 'tmpl-test-com',
    
    data: function() {
      return {
        localizeData: []
      }
    },
    
    methods: {
      setLocalData: function() {
          var self = this;
          
          data : {
            action: 'test-set-local-data'
          };
          
          jQuery.post( NinjaTodo.ajaxurl, data, function(resp) {
               if ( resp.success ) {
                  self.localizeData = resp.data;
               }
          });
        
      
      }
    },
    
    created: function() {
      this.setLocalData();
    }
});

// ----------------- PHP file 

add_action( 'wp_ajax_test-set-local-data', 'set_localize_data' );

function set_localize_data() {
  return array(
    'save' => __( 'Save', 'ninja-todo' ),
    'add_new' => __( 'Add New', 'ninja-todo' ),
    'cancel' => __( 'Cancel', 'ninja-todo' )
  );
}

// ---------------- .vue file 

<div class="test-class">
    <input type="button" name="save" value="{{ localizeData.save }}" >
    <input type="button" name="add_new_btn" value="{{ localizeData.add_new }}" >
    <input type="button" name="cancel_btn" value="{{ localizeData.cancel }}" >
</div>