Yes ,cordova can write into the app folder or in anyfile which it Android can access , if you want to write in private other folder that wont be possible .
For cordova you need to just add a plugin for accessing the storage or create a custom plugin for the same.
Below is code for writing into your App Folder :
if (!Environment.getExternalStorageState().equals(Environment.MEDIA_MOUNTED)){
Log.d("Log", "No sdcard available");
} else {
File directory = new File(Environment.getExternalStorageDirectory()+File.separator+"ApplicationFolderName");
directory.mkdirs();
}
Make sure you add permission into manifest file also.