@kado897 For some reason, the domain name the link points to expired, and the link just shows one of those useless placeholder "search" pages. So, antivirus and firewall blocking probably didn't apply when I had the problem, especially since it happened to a lot of other pages also.
Anyway, I believed I solved the problem, but a while after creating this thread, so I forgot to mark it as solved. The problem lies in the fact that Flash requires admin rights to install (like most programs), but when it modifies the registry to allow IE to use the plugin, the registry information is owned by the administrator, and can't be read by a regular user. Basically, the information that should be telling IE to use the Flash plugin is restricted to admin users only.
The solution lies here: windows 7 - Embedded video is not shown in Internet Explorer 9 RC or Beta - Super User
. It involves editing the registry as an administrator, so be sure to make a backup first. I also think it needs to be done every time Flash updates. Of course, a more permanent solution would be to use another browser like Firefox or Chrome... I tried Firefox and it worked even when IE9 didn't.