Here's a quick solution: in short, you will need to install webpack, which is a module bundler (it means that its main purpose is to bundle JavaScript files for usage in a browser). If you have npm already set up, you can execute this command in your project:

npm install webpack

After you have done that you can proceed to set up firebase (which, from what I can see from your image, you have already done). You will need to run another command:

npm install firebase

Continuing the setup of webpack, you will need to create a webpack.config.js file and there set the entry and the output. Again, you can find plenty of tutorials online, but here's a quick example implementation:

webpack.config.js:

const CopyWebpackPlugin = require('copy-webpack-plugin');
const webpack = require('webpack');
const path = require('path');

module.exports = {
    mode: 'production',
    entry: {
        main: './src/main'
    },
    output: {
        filename: '[name].bundle.js',
        path: path.resolve(__dirname, 'dist')
    },
    module: {
        rules: [
            {
                test: /\.js$/,
                exclude: /node_modules/
            },
            {
                test: /\.(png|svg|jpg|gif)$/,
                use: [
                    'file-loader',
                ],
            },
        ],
    },
    devServer: {
        contentBase: './dist',
        overlay: true,
        hot: true
    },
    plugins: [
        new CopyWebpackPlugin({
            patterns: [
                { from: 'manifest.json', to: 'manifest.json' },
            ],
        }),
        new CopyWebpackPlugin({
            patterns: [
                { from: 'images', to: 'images' },
            ],
        }),
        new CopyWebpackPlugin({
            patterns: [
                { from: 'popup.html', to: 'popup.html' },
            ],
        }),
        new webpack.HotModuleReplacementPlugin()
    ],


};

Once you've done that, in your entry file (the entry point), you can import firebase and set it up:

main.js

 import { initializeApp } from 'firebase/app';
    
// TODO: Replace the following with your app's Firebase project configuration
const firebaseConfig = {
  //...
};

const app = initializeApp(firebaseConfig);

When you run npm start, webpack will create another folder (the 'dist' folder). This folder is the chrome exstension with firebase set up!

Hope I was able to help you. If you have any questions feel free to ask!