My Flex application runs in a browser, not AIR, but the concept is the same; it connects to an Oracle database, via two different server-side technologies (I've been experimenting for my employer) : Ruby-on-Rails or PL/SQL (using the Oracle HTTP Server and mod_plsql).
Basically, your Flex application can talk to any database, as long as you have a server-side service which dialogs with it. I use XML to send data back and forth between the server and the client (browser). Cheers, Chris.
