Unless I 've misunderstood something...
...older websites which were written in VB and use Access databases, and are currently orphaned because the Access to SQL Server migration is non-trivial, can, or so it appears from a quick documentation scan, be migrated relatively easily by running their code under Mono. Mono uses ODBC to talk to databases including (by implication) at least some flavours of SQL Server. Given that the ODBC client API is meant to be RDBMS-agnostic, it should follow that, once your code is running under Mono it should be simple to move it to PostgreSQL by migrating the data and then swapping the SQLServer ODBC module for the PostgreSQL one.
But, I probably missed something during that quick scan, so would appreciate knowing what I missed.
I've written systems using various databases including Sybase (SQL Server's parent), Access, Oracle and PostgreSQL and used both ODBC and JDBC but have only given Mono and its VB support a fairly quick preliminary scan to see whether the approach outlined above might offer an easier migration path for a VB+access website than migration to SQL Server. I already know that Mono can talk to an Apache web-server.