After some Googling, I came across this forum post: http://social.microsoft.com/Forums/en-US/crm/thread/b86740b6-6418-4e1c-9020-1d6c9c630b7b/
Basically a hotfix KB956391 broke the client control for SSRS. We had already installed all the latest automatic updates on all the servers, but the following update isn’t presented with Windows Update.
Installing this fix on ALL* related servers, then rebooting fixed the problem. (*You can’t just install it on the SSRS machine, you need to install it on any server that allows a client to connect to reporting services.)
Basically, this fix forces the browser to download the latest client viewer and install it. I had found a number of other solutions that required a manual installation on the client’s computer, but this one works from the server side.