docs: Add Flutter code sample for Firebase Auth integration guide (#31087)

* docs: Add Flutter code sample for accessToken function

* docs: fix the code sample

apps/docs/content/guides/auth/third-party/firebase-auth.mdx

@@ -36,6 +36,26 @@ Make sure the all users in your application have the `role: 'authenticated'` [cu
 
 </TabPanel>
 
+<TabPanel id="dart" label="Flutter">
+
+Creating a client for the Web is as easy as passing the `accessToken` async function. This function should [return the Firebase Auth JWT of the current user](https://firebase.google.com/docs/auth/admin/verify-id-tokens) (or null if no such user) is found.
+
+```dart
+await Supabase.initialize(
+  url: supabaseUrl,
+  anonKey: supabaseKey,
+  debug: false,
+  accessToken: () async {
+    final token = await FirebaseAuth.instance.currentUser?.getIdToken();
+    return token;
+  },
+);
+```
+
+Make sure the all users in your application have the `role: 'authenticated'` [custom claim](https://firebase.google.com/docs/auth/admin/custom-claims) set. If you're using the `onCreate` Cloud Function to add this custom claim to newly signed up users, you will need to call `getIdToken(/* forceRefresh */ true)` immediately after sign up as the `onCreate` function does not run synchronously.
+
+</TabPanel>
+
 <TabPanel id="swift" label="Swift (iOS)">
 
 ```swift