Welcome Aboard
Access 2003 cannot talk to an .accdb so if you need to automate the export from that perspective, you need to export to a csv, txt, or xls file or an intermediate .mdb.
Then the A2010 application will need to import from the intermediate csv,txt,xls, or mdb file.
I don't have MS-Access 2003, but both in MS-Access 2000 and MS-Access 2010, the following would do it.
Mark the table to export in the database - right click, choose export.