In the simplest form as you have described in your question, saying that you are just want to display content from the current user in a module, it is possible that some modules would have already such filter settings (that will use the current user id and retrieve/display their articles in the front-end). You can look in JED for article modules and check for their features.
A module that provides plenty of filtering features, is the RAXO Pro module (commercial). It has an author filtering option to display : My articles or not my articles, that possibly might do the trick (but better ask a presale question).
If you won't find other modules, then you could customize a little the Articles Category module, that provides author filtering, and set it to use the current user id as the author.
Going deeper, you could create your own custom menu item for the com_content component and set up such filtering.
However, based on your comment above, I think that you are going to need a much more sophisticated solution, than a simple module or "iframe". There are various extensions that deal with the front-end article submission/authoring. Check the relevant JED category to get some ideas.
I have seen extensions that deal exactly with this situation in the past, but I can't recall their names now.
Some other ideas would be to create your own content items with a CCK e.g K2, or a custom app with an application builder like Fabrik, Seblod or other similar tools.
To add some extra info, in the case you want to create your custom page, module, the key here is to use the current user id, as well as to check for the user access (probably users should be logged in, in order to submit articles or to see any unpublished articles).
To get the user id:
$user = JFactory::getUser();
Then you can display articles created by the user with this id.
Lastly, you should also set your desired viewing access level for that page/module (e.g. registered), so it will prompt user to login, before being able to see the content.