Feedback form between Admin and Employees

What's wrong with one of available Issue Tracking Systems? Bugzilla, Trac?

See:
http://en.wikipedia.org/wiki/Issue_tracking_system[^],
http://en.wikipedia.org/wiki/Comparison_of_issue_tracking_systems[^],
http://en.wikipedia.org/wiki/Issue_management[^].

All your requirements can be well observed.

[EDIT: answering after clarification]

You should have a regular Web form with the method "POST". Look how my own form works: http://www.sakryukov.org/contact/[^].

You validate the form on server side and provide further action on post using the posted data, for example, send e-mail internal to host.
Warning! Be very careful! You should not send e-mail directly. You should validate all data, especially the data which you may want to use in e-mail headers. For a malicious user, it is ridiculously simple to turn the host in a zombie re-sending spam or something like that. Make sure that there is nothing unusual in data, especially in the address, subject line, etc. In particular, make sure there is no any non-printed characters like '\n', '\r', etc. — Web user cannot enter them, but malicious user can easily insert them programmatically. Filter everything unexpected and report an attempt of malicious activity internally.

—SA