I figured it out; I found some forum posts that mentioned I had to update my Manifest file. Apparently Oracle recently changed the way Liveconnect works so you have to be explicit about allowing subdomains that can call the applet.
The information on that is here, and basically you have to add the Caller-Allowable-Codebase
directive to the Manifest file. Fortunately it can be set to a wildcard, which is what I ended up doing. Apparently you're also supposed to set Application-Name
along with it in the manifest file, and you're specifically NOT supposed to include Trusted-Library
true. I don't really know why or what that does, but I was able to adjust my manifest file and everything seems to work again!