Is there a way to create a Personal Geodatabase in QGIS?

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 middle-ware.

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.

QGIS uses the OGR library for the majority of it's GIS format access. The Personal GDB access falls under this grouping.

Based on the OGR Vector Format's page here: OGR Vector Formats

The ESRI Personal GeoDatabase driver does not have Creation support.

I had the same question now - what alternative to 'geodatabase' is there in QGIS that could hold multiple layers with attribute tables in a single file. I did not find answer at first, but I found this question.

After some research I found that SpatiaLite database is nice replacement for it tutorial http://www.qgistutorials.com/en/docs/digitizing_basics.html

Also I found that SpatiaLite is transferable between ArcGis and QGIS see this blog: http://blog.geomusings.com/2013/08/07/spatialite-and-arcgis-10-dot-2/