You can use my plugin flutter_inappwebview, which is a Flutter plugin that allows you to add inline WebViews or open an in-app browser window and has a lot of events, methods, and options to control WebViews.

localStorage feature is enabled by default!

Here is a quick example that sets and retrieve a localStorage value when the page stops loading:

import 'dart:async';
import 'package:flutter/material.dart';
import 'package:flutter_inappwebview/flutter_inappwebview.dart';

Future main() async {
  WidgetsFlutterBinding.ensureInitialized();
  runApp(MyApp());
}

class MyApp extends StatefulWidget {
  @override
  _MyAppState createState() => new _MyAppState();
}

class _MyAppState extends State<MyApp> {
  InAppWebViewController _webViewController;

  @override
  void initState() {
    super.initState();
  }

  @override
  void dispose() {
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        appBar: AppBar(
          title: const Text('InAppWebView Example'),
        ),
        body: Container(
            child: Column(children: <Widget>[
              Expanded(
                  child: InAppWebView(
                    initialUrl: "https://github.com/flutter",
                    initialHeaders: {},
                    initialOptions: InAppWebViewGroupOptions(
                      crossPlatform: InAppWebViewOptions(
                        debuggingEnabled: true,
                      ),
                    ),
                    onWebViewCreated: (InAppWebViewController controller) {
                      _webViewController = controller;
                    },
                    onLoadStart: (InAppWebViewController controller, String url) {

                    },
                    onLoadStop: (InAppWebViewController controller, String url) async {
                      await controller.evaluateJavascript(source: "window.localStorage.setItem('key', 'localStorage value!')");
                      await controller.evaluateJavascript(source: "alert(window.localStorage.getItem('key'))");
                    },
                  ))
            ])),
      ),
    );
  }
}

Screenshot: