Altra conversione in Typescript , partendo da un progetto di esempio funzionante...

src/views/Auth.vue

@@ -0,0 +1,24 @@
+<template lang="html">
+  <div id="firebaseui-auth-container"></div>
+</template>
+
+<script>
+  export default {
+    name: 'auth',
+    mounted () {
+      var uiConfig = {
+        signInSuccessUrl: '/success',
+        signInOptions: [
+          //firebase.auth.GoogleAuthProvider.PROVIDER_ID,
+          //firebase.auth.EmailAuthProvider.PROVIDER_ID
+        ]
+      }
+      //var ui = new firebaseui.auth.AuthUI(firebase.auth())
+      //ui.start('#firebaseui-auth-container', uiConfig)
+    }
+  }
+</script>
+
+<style lang="css">
+
+</style>