Classic ASP can connect with a variety of databases.

Microsoft Access databases are surprisingly good for running small to medium sized websites. I've had one grow to almost 1GB before causing any issues. They have security benefits over more complex databases being almost immune to SQL injection and they're portable - great for backing up or moving your site. However not all shared web hosts will support them.

MySQL databases are available on pretty much all web hosting, and normally have a control panel allowing you to manage the structure and data.

MS SQL (SQL Server) databases are Microsoft's grown-up DBs. A lot more complex to use and set up they can handle many GBs of data even on the free Express version.

Each of these have their own idiosyncrasies when it comes to using SQL to read and update data. I have lots of experience in moving sites from using one type of DB to another.

If you find your website can read (select) from an Access database but can't update or delete anything in it, it's most likely to be a permissions problem on the web hosting - maybe your host has moved you to a new server without correctly carrying the permissions over on the folder the database is stored in. But also, check your connection string - it's possible to configure this in a way which means it's read-only.