If this isn't possible what would be the closest equivalent open source? Could parts of the module be ported to work on 9.3 or QGIS? I'm not sure if this would infringe on the license porting for 9.3.
For ArcGIS 9.3, no because arcpy requires ArcGIS 10 geoprocessor to be installed.
Python is heavily used in QGIS - but you would have to replicate (program) QGIS python to emulate any ArcGIS functionality.
your best bet is the QGIS API http://doc.qgis.org/stable/
(GISse user Underdark has much fuller knowledge and understanding of this see their work http://users.qgis.org/planet/user/4/)
It should be possible to access arcpy from a qgis on a machine that also has both qgis and arcgis, see How do I access arcobjects from python?, and Can I use python 2.6 with ArcGIS 9.3? for how to maintain concurrent python installs.
I think what you're really after though is "how do I do arcgis-like things in python without arcgis?". For this, as Underdark said, you're going to need to break your question down into specific tasks/objectives, and ask as individual questions.