The problem is that Word stores the full path name of the data source. When
you move the files to another machine, the full pathname will be different
because the user's name is in the full path, e.g.

c:\Documents and Settings\myname\My Documents\My Data Sources\mydata.mdb

There are two things you could do:
a. put the data in a file whose pathname does not vary from machine to
machine or user to user. e.g. c:\mergedata\mydata.mdb

or perhaps somewhere under

c:\Documents and Settings\All Users\

b. disconnect the data source before you move the files (by changing the
mail merge main document back to a normal document). You will lose any
sort/filter settings. You can either explan to the user how to reconnect to
the data, or provide a VBA AutoOpen macro that creates the correct path name
of the data file and uses OpenDataSource to connect to it.