If you start with Access as your forum grows you should upsize the database to SQL Server which is also an RDMS.
We have Web Wiz Forums running on our network with as many as 5 million posts using SQL Server, however we find those that use the mySQL version we often have to move to SQL Server due to database and performance issues at around 300,000 posts.
The myODBC driver required for ASP to talk to a mySQL database is also very buggy, we are forever trying to find work arounds to error caused by this driver. It causes all sorts of stability issues.
Access is fine as a desktop database, mySQL is fine with PHP, but with ASP you should be looking at using SQL Server.
Most web hosts, like ourselves, offer SQL Server as standard with all hosting plans, so it is not that expensive these days to use SQL Server.