你可以试试我的插件flutter_inappwebview。
要请求有关摄像头和麦克风的权限，您可以使用permission_handler插件。
此外，它还有Android的androidOnPermissionRequest事件，这是当WebView请求访问指定资源的权限时触发的事件（即Android原生WebChromeClient.onPermissionRequest事件）。
在Android上使用WebRTC的示例：

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

Future main() async {
  WidgetsFlutterBinding.ensureInitialized();

  await Permission.camera.request();
  await Permission.microphone.request();

  runApp(MyApp());
}

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

class _MyAppState extends State<MyApp> {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
        home: InAppWebViewPage()
    );
  }
}

class InAppWebViewPage extends StatefulWidget {
  @override
  _InAppWebViewPageState createState() => new _InAppWebViewPageState();
}

class _InAppWebViewPageState extends State<InAppWebViewPage> {
  InAppWebViewController _webViewController;

  @override
  Widget build(BuildContext context) {
    return Scaffold(
        appBar: AppBar(
            title: Text("InAppWebView")
        ),
        body: Container(
            child: Column(children: <Widget>[
              Expanded(
                child: Container(
                  child: InAppWebView(
                      initialUrl: "https://appr.tc/r/158489234",
                      initialOptions: InAppWebViewGroupOptions(
                        crossPlatform: InAppWebViewOptions(
                          mediaPlaybackRequiresUserGesture: false,
                          debuggingEnabled: true,
                        ),
                      ),
                      onWebViewCreated: (InAppWebViewController controller) {
                        _webViewController = controller;
                      },
                      androidOnPermissionRequest: (InAppWebViewController controller, String origin, List<String> resources) async {
                        return PermissionRequestResponse(resources: resources, action: PermissionRequestResponseAction.GRANT);
                      }
                  ),
                ),
              ),
            ]))
    );
  }
}

此示例使用https://appr.tc/上的房间158489234，这是一个基于WebRTC（https://github.com/webrtc/apprtc）的视频聊天演示应用程序。要使其工作，您需要将选项mediaPlaybackRequiresUserGesture设置为false。
另外，您需要在AndroidManifest.xml中添加以下权限：

<uses-permission android:name="android.permission.INTERNET"/>
<uses-permission android:name="android.permission.CAMERA" />
<uses-permission android:name="android.permission.RECORD_AUDIO" />
<uses-permission android:name="android.permission.MODIFY_AUDIO_SETTINGS" />
<uses-permission android:name="android.permission.VIDEO_CAPTURE" />
<uses-permission android:name="android.permission.AUDIO_CAPTURE" />

import 'package:flutter/foundation.dart';
import 'package:flutter/material.dart';
import 'package:flutter_inappwebview/flutter_inappwebview.dart';
import 'package:permission_handler/permission_handler.dart';

Future main() async {
  WidgetsFlutterBinding.ensureInitialized();
  if (!kIsWeb &&
      kDebugMode &&
      defaultTargetPlatform == TargetPlatform.android) {
    await InAppWebViewController.setWebContentsDebuggingEnabled(kDebugMode);
  }

  await Permission.camera.request();
  await Permission.microphone.request(); // if you need microphone permission

  runApp(const MaterialApp(home: MyApp()));
}

class MyApp extends StatefulWidget {
  const MyApp({Key? key}) : super(key: key);

  @override
  State<MyApp> createState() => _MyAppState();
}

class _MyAppState extends State<MyApp> {
  final GlobalKey webViewKey = GlobalKey();

  InAppWebViewController? webViewController;

  @override
  Widget build(BuildContext context) {
    return Scaffold(
        appBar: AppBar(
          title: const Text("InAppWebView test"),
        ),
        body: Column(children: <Widget>[
          Expanded(
            child: InAppWebView(
              key: webViewKey,
              initialUrlRequest: URLRequest(url: Uri.parse("https://webrtc.github.io/samples/src/content/getusermedia/gum/")),
              initialOptions: InAppWebViewGroupOptions(
                crossPlatform: InAppWebViewOptions(
                  mediaPlaybackRequiresUserGesture: false,
                ),
              ),
              onWebViewCreated: (controller) {
                webViewController = controller;
              },
            ),
          ),
        ]));
  }
}