We are trying to setup a shipment of medical supplies to a hospital or
clinic in Iraq.  We have a great organization to go through
(www.medisend.org), but to get this started, we need to find a
reliable contact at a hospital in Iraq.   Any ideas?

Have you thought of going through another organization?  Medecins Sans
Frontieres, an international humanitarian aid organisation that
provides emergency medical assistance to populations in danger,
currently has medical teams working there.
http://www.msf.org/content/page.cfm?articleid=1AC93676-285D-4C66-900940BCB297B101

Dear Max, 

I agree with Tehuti. There are other organisations you might want to
contact, like the Red Crescent:
http://www.ifrc.org/WHAT/disasters/response/iraq.asp

CARE work in hospitals in Iraq :
http://www.care.ca/press/pr/iraq/030417_e.shtm

Google Directory has a variety of organisations who work in Iraq: 
http://directory.google.com/Top/Regional/Middle_East/Iraq/Society_and_Culture/Humanitarian_Issues_and_Aid/

In this transtional stage it might be difficult to organise a direct
shipment to a hospital or clinic.