I don't know if it is possible to add a Flash banner into a Wordpress theme... Ideally, this banner would be in the header of the site always. Any ideas on how I might accomplish this?
You need to edit the corresponding .php file that contains your banner (I don't know what version of WordPress you're using).
The code will be resource-intensive and ugly, though.
Yes, it's very much possible. What I would recommend is to use swfobject to replace the your current header with a Flash movie. This way it will degrade gracefully for everyone that lacks a proper flash player, and also you get the benefit of proper google indexing.