Flutterはまだ慣れておらず特にUIや関数でない部分、iOSやAndroidフォルダ側の独自の設定で戸惑うことがあるためメモ的に残しておきます。

info.plistにタグを追加

ios/Runnerフォルダにある環境設定ファイルであるinfo.plistにXMLを追加します。

info.plist

<dict>
    ...省略
	<key>NSMicrophoneUsageDescription</key>
	<string>Used to capture audio for image picker plugin</string>
    ...省略
</dict>

をdictタグ内の適当な場所に追加。

Podfileの編集

iOS,macOSのライブラリ管理ツールCocoaPodsの設定ファイルであるPodfileを書き換えてmicrophoneの設定を追記します。

post_install do |installer|
  installer.pods_project.targets.each do |target|
    flutter_additional_ios_build_settings(target)
  end
end

を下に置き換え

post_install do |installer|
  installer.pods_project.targets.each do |target|
    flutter_additional_ios_build_settings(target)

    target.build_configurations.each do |config|
      # You can remove unused permissions here
      # for more information: https://github.com/BaseflowIT/flutter-permission-handler/blob/master/permission_handler/ios/Classes/PermissionHandlerEnums.h
      # e.g. when you don't need camera permission, just add 'PERMISSION_CAMERA=0'
      config.build_settings['GCC_PREPROCESSOR_DEFINITIONS'] ||= [
        '$(inherited)',

        ## dart: PermissionGroup.calendar
        # 'PERMISSION_EVENTS=1',

        ## dart: PermissionGroup.reminders
        # 'PERMISSION_REMINDERS=1',

        ## dart: PermissionGroup.contacts
        # 'PERMISSION_CONTACTS=1',

        ## dart: PermissionGroup.camera
        # 'PERMISSION_CAMERA=1',

        ## dart: PermissionGroup.microphone
        'PERMISSION_MICROPHONE=1',

        ## dart: PermissionGroup.speech
        # 'PERMISSION_SPEECH_RECOGNIZER=1',

        ## dart: PermissionGroup.photos
        # 'PERMISSION_PHOTOS=1',

        ## dart: [PermissionGroup.location, PermissionGroup.locationAlways, PermissionGroup.locationWhenInUse]
        # 'PERMISSION_LOCATION=1',

        ## dart: PermissionGroup.notification
        # 'PERMISSION_NOTIFICATIONS=1',

        ## dart: PermissionGroup.mediaLibrary
        # 'PERMISSION_MEDIA_LIBRARY=1',

        ## dart: PermissionGroup.sensors
        # 'PERMISSION_SENSORS=1',

        ## dart: PermissionGroup.bluetooth
        # 'PERMISSION_BLUETOOTH=1',

        ## dart: PermissionGroup.appTrackingTransparency
        # 'PERMISSION_APP_TRACKING_TRANSPARENCY=1',

        ## dart: PermissionGroup.criticalAlerts
        # 'PERMISSION_CRITICAL_ALERTS=1',
      ]

    end
  end
end

'PERMISSION_MICROPHONE=1'のみコメントをインさせています。他に必要なパーミッションがあれば任意で設定。

その後

pod upgrade
pod install

で設定完了