Hamad Medical Centre runs the three main Hospitals here and their website’s ‘vacancies’ page is “under instruction�. However, if you go to Human Resources on http://www.hmc.org.qa/hr/jvoc.htm, you’ll get some useful information on housing, etc, and an email address to apply to. I gather staff turn-over is high so vacancies will frequently arise. Most of the nursing staff are Indian and Philippina while doctors tend to be Northern Arab.
The American Hospital doesn’t seem to have its own website.
Good luck!