Is it possible to monitor what is happening to an Access MDB (ie. what SQL queries are being executed against it), in the same way as you would use SQL Profiler for the SQL
The answer depend on the technology used from the client which use MDB. There are different tracing settings which you can configure in HKEY_LOCAL_MACHINE\Software\Microsoft\Jet\4.0\Engines\ODBC http://office.microsoft.com/en-us/access/HP010321641033.aspx. If you use OLEDB to access MDB from SQL Server you can use DBCC TRACEON (see http://msdn.microsoft.com/en-us/library/ms187329.aspx). I can continue, but before all you should exactly define which interface you use to access MDB.
MDB is a file without any active components, so the tracing can makes not MDB itself, but the DB interface only.
UPDATED: Because use use DAO (Jet Engine) and OLE DB from VB I recommend you create JETSHOWPLAN regisry key with the "ON" value under HKEY_LOCAL_MACHINE\SOFTWARE\MICROSOFT\JET\4.0\Engines\Debug (Debug subkey you have to create). This key described for example in https://web.archive.org/web/1/http://articles.techrepublic%2ecom%2ecom/5100-10878_11-5064388.html, http://msdn.microsoft.com/en-us/library/aa188211%28office.10%29.aspx and corresponds to http://support.microsoft.com/kb/252883/en allow trace OLE DB queries. If this output will be not enough for you you can additionally use TraceSQLMode and TraceODBCAPI from HKEY_LOCAL_MACHINE\Software\Microsoft\Jet\4.0\Engines\ODBC. In my practice JETSHOWPLAN gives perfect information for me. See also SHOWPLAN commend.
UPDATED 2: For more recent version of Access (like Access 2007) use key like HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Office\12.0\Access Connectivity Engine\Engines. The tool ShowplanCapturer (see http://www.mosstools.de/index.php?option=com_content&view=article&id=54&Item%20%20id=57, to download http://www.mosstools.de/download/showplan_v9.zip also in english) can be also helpful for you.