Database triggers unfortunately are not an option (as they are related to the database table and can only do in-database operations). However you could write some PHP triggers. These would essentially need to be inserted in front or behind the current MySQL database inserts, to do the MSAccess inserts identically.
This is probably the way I would consider first. Unfortunately I don't think there is any easy option.
Possibly another option would be to create a trigger on the MySQL table that you want to copy that will dump a copy of the table to a CSV file/directory. After which you could periodically run another program outside of your web application to read this directory CSV entry and update the MSAccess DB.
Question: are you copying entrys both ways, i.e. MSAccess -> MySQL && MySQL-> MSAccess .
The reason I ask is because you have to be very careful you're only adding in new entrys to each and not overwriting old ones, or removing newly inserted data. Also, if you have auto incrementing primary keys, this could cause a duplication problem.
Have you considered making your vb6 app use MySQL instead? Then they can both use the same database and updates on one will be automatically shown in the other application.