Вы можете использовать пакет audio_recorder:
https://pub.dev/packages/audio_recorder
// Import package
import 'package:audio_recorder/audio_recorder.dart';
// Check permissions before starting
bool hasPermissions = await AudioRecorder.hasPermissions;
// Get the state of the recorder
bool isRecording = await AudioRecorder.isRecording;
// Start recording
await AudioRecorder.start(path: _controller.text, audioOutputFormat: AudioOutputFormat.AAC);
// Stop recording
Recording recording = await AudioRecorder.stop();
print("Path : ${recording.path}, Format : ${recording.audioOutputFormat}, Duration : ${recording.duration}, Extension : ${recording.extension},");