Is there an issue reporting module with these features? We have 3 departments and we give IT support for all project users. We would like to receive there support request using our drupal 7 site  instead of receiving phone call. 


We just need a support request form
The form sends a notification email to experts  
The submitted issues should be listed by project group; only experts can access the list. 
The form access privilege is only for authenticated users.
There is no progress status report for the users.


What do you suggest?
The Webform module might do what you want. It lets you create forms using the user interface. You can configure which actions happen when a user submits a form, such as email one or more users and store it in a table.

Update about your comment, which looks like so:


  Webform seems good but i drop it by one important reason. that is it doesn't give a privilege form submission only the authenticated user!


Forms created with Webform are just a type of node. If you want to restrict access to the forms you create, you can either not allow anonymous to access any node or you can use more fine grain permissions with another module like Nodeaccess. If this is for an intranet site, the first option might be the best