In both cases, Server-side and Client-side, the Access mdb file resides on the server. As a file, its just a bunch of bits.
With server-side PHP, the toolset to correctly access the Access mdb file, either ADO or ODBC, would run on the web server. I am not aware of these being available to run under Linux.
With client-side JavaScript, the toolset to correctly access the Access mdb file, either ADO or ODBC, would run on the client machine (you or your boss's desktop machine).
In either case the Access.exe application program does not run; only the mdb file is used.
PHP/MySQL is still the best method.