According to the GDAL docs:
OGR optionally supports reading ESRI Personal GeoDatabase .mdb files
via ODBC. Personal GeoDatabase is a Microsoft Access database with a
set of tables defined by ESRI for holding geodatabase metadata, and
with geometry for features held in a BLOB column in a custom format
(essentially Shapefile geometry fragments). This drivers accesses the
personal geodatabase via ODBC but does not depend on any ESRI
Writing a personal geodatabase on the other hand, is another story. Access mdb is a proprietary file format and open source projects like QGIS tend to stay away from them. I did see an old open source project called MDB Tools which aims to
MDB Tools is an open source suite of libraries and utilities to read
(and soon write) MDB database files.
I'm not sure how far along their goal that project is though. So no, I don't think you can create personal geodatabase files using QGIS as of now or any time soon.
As for your client's request, @Ragi has just finished writing an ArcGIS plugin that let's you use OGR sources, this includes Spatialite and PostGIS. That way you can use ArcGIS and PostGIS ( or Spatialite, whichever you prefer ). You might want to give it a try.
I hope that helps.