I wrote an Applet which writes in a MS Access Database using ODBC.

From Java, you shouldn't be speaking ODBC directly. What you should be using is JDBC. If you happen to be using the JDBC-ODBC bridge driver that's fine, but your program was using JDBC...

To read/write from/to an MS Access DB from a Mac, get a Type 3/4 driver for Access, instead of using the bridge driver.