How to use Github Pages with Dependencies?

it fixes the issue No. It runs Jekyll, and only Jekyll, and only supports the plugins contained in the gh-pages gem. Bower is run on node.js. So the only way to do this is to resolve the dependencies locally, commit everything and then push.
In order to keep your branches somewhat clean, you could use a source (or whatver name you prefer) branch which only contains the needed sources and make your changes there. Once you have done your changes, merge the source branch into master, then resolve the dependencies again (if you changed them, or updates are available), commit and push the master branch.

I wish did fix the issue. It depends from where you want to get the dependencies.
Do you want to host them yourself on your own site? Maybe it's not necessary. Instead you could get them directly from their original CDN.

<script src="https://unpkg.com/@webcomponents/webcomponentsjs@^2/webcomponents-bundle.js"></script>

import {html, render} from 'https://unpkg.com/lit-html?module'

help you fix your problem Nothing can be executed on the server side on GitHub, beside GitHub Actions.
That means GitHub can only host the sources for your project, but cannot be a release/deployment destination.

it helps some times This is happening because of your use of the GITHUB_TOKEN variable. There's an open issue with GitHub due to the fact that the built in token doesn't trigger the GitHub Pages deploy job. This means you'll see the files get committed correctly, but they won't be visible.
To get around this you can use a GitHub access token. You can learn how to generate one here. It needs to be correctly scoped so it has permission to push to a public repository. You'd store this token in your repository's Settings > Secrets menu (Call it something like ACCESS_TOKEN), and then reference it in your configuration like so:

on:
push:
    branches:
    - master

jobs:
build:

    runs-on: ubuntu-latest

    steps:
    - uses: actions/checkout@v1
    - name: Deploy with github-pages
    uses: JamesIves/github-pages-deploy-action@master
    env:
        ACCESS_TOKEN: ${{ secrets.ACCESS_TOKEN }}
        BASE_BRANCH: master # The branch the action should deploy from.
        BRANCH: gh-pages # The branch the action should deploy to.
        FOLDER: target # The folder the action should deploy.
        BUILD_SCRIPT: npm install && npm run-script build

wish of those help Delete you Gemfile.lock and run a bundle update command.
You will get up to date dependencies.