Just to be explicit: I'm led to believe that the security of data is primarily centered around securing the web server and avoiding use of certain code that would allow exploits (e.g. SQL Injections) and less to do with physical storage of data. In other words, many web server communicate with database server in relatively unsecured manner because they're physically secured by say, network firewalls.
The real problem with using Access as a data storage for a website is that it isn't really designed to be used as a service and thus may have issues. Since there's free RDBMSes out there to pick from, there's no reason to use Access. Besides, just because you use one of those free RDBMSes, doesn't mean you can't use Access with the same RDBMS - you can if you need a rich front-end for maybe internal use while exposing same data via web for different consumptions.