Add a CI check for `web/samples_index` (#1827)

fixes: https://github.com/flutter/samples/issues/1825

Note: the added check is expected to fail until
https://github.com/flutter/samples/pull/1826 is landed

## Pre-launch Checklist

- [x] I read the [Flutter Style Guide] _recently_, and have followed its
advice.
- [x] I signed the [CLA].
- [x] I read the [Contributors Guide].
- [x] I updated/added relevant documentation (doc comments with `///`).
- [ ] All existing and new tests are passing.

If you need help, consider asking for advice on the #hackers-devrel
channel on [Discord].

<!-- Links -->
[Flutter Style Guide]:
https://github.com/flutter/flutter/wiki/Style-guide-for-Flutter-repo
[CLA]: https://cla.developers.google.com/
[Discord]: https://github.com/flutter/flutter/wiki/Chat
[Contributors Guide]:
https://github.com/flutter/samples/blob/main/CONTRIBUTING.md

---------

Co-authored-by: Parker Lougheed <parlough@gmail.com>

.github/workflows/main.yml

@@ -36,6 +36,16 @@ jobs:
         with:
           channel: ${{ matrix.flutter_version }}
       - run: ./tool/flutter_ci_script_${{ matrix.flutter_version }}.sh
+  web-samples-index:
+    name: web/samples_index config check
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@8e5e7e5ab8b370d6c329ec480221332ada57f0ab
+      - uses: subosito/flutter-action@48cafc24713cca54bbe03cdc3a423187d413aafa
+      - run: |
+          dart pub get
+          dart run grinder generate
+        working-directory: web/samples_index
   android-build:
     runs-on: ubuntu-latest
     if: github.repository == 'flutter/samples'