This issue 'resolved itself', possibly because fedex changed the coding on their website to work more reliably with chrome. Or else chrome quietly updated and is now more compatible with the fedex site's print function.
There ws a separate issue that may be related, for a while the Zebra label printer stopped working with newer versions of Java, so if you updated Java when prompted, you had to go to the Java website, download an older build, uninstall the new build, then install the older one. Fedex tech support walked me through this.
As of around 8/1/2014 or so, this issue was fixed by Fedex, and now the label printer works with the newest version of Java (In my case Version 7, Update 67). So feel free to update Java and use chrome with the fedex site.
In fact, I now have an ongoing issue where Firefox won't work with it because Java simply doesn't appear in the list of installed firefox plugins. I guess I'll research that but if anyone has struggled with both chrome and firefox, and they eventually got a modern version of Firefox working, I'd appreciate any tips.