Hi, I think you will be needing to read the OutLook API to recieve the emails on your website. Otherwise there is no such built-in feature that will allow you to get the OutLook emails on your website.
You need to use API, pass some credentials and then get the result out HTTPS and show them on the web page.
However, to deal with the ASP.NET and send the emails, attach files etc I have written an article feel free to check it out.
Sending emails easily using ASP.NET helpers
But remember, you will be needed to read the Outlook API to integrate their inbox to your own website's inbox.