I'm an Access promoter, but not for use on websites because Jet/ACE is not threadsafe (though Michael Kaplan once said that is is threadsafe if you access it via ADO/OLEDB; I don't quite understand how a database abstraction layer can wash away a characteristic of the underlying database engine it's calling, but if MichKa said, it's 99% likely to be true).
Now, the exceptions would be if you're using it for prototyping something that will use a different database, or if it's read-only, or is read-write but will only ever have a very small number of users.
Michael Kaplan's website, trigeminal.com, used to use a Jet database as the back end (it may still -- I don't know that MichKa ever changed it), and when that was his main website he reported getting 100K hits a day. But it's a read-only site, so fits my restrictions.
There are so many different alternatives and they are mostly easy to use that I just don't see the point of trying to use Jet/ACE as back end for a website. I'd never do it myself (all the websites I'm responsible for use MySQL).