C:\Windows\System32\config\systemprofile is treated as your HOME doirectory for some reason... http://www.coderanch.com/t/517573/java/java/System-getProperty-user-home-unreliable looks related. But this is a system folder you can't write to - thus the problem.
I'd suggest passing a different user home to WebStorm - just add -Duser.home=<your preferred location> to bin/WebStorm.exe.vmoptions file
Also, the config/system files location can be explicitly changed in bin/idea.properties file, idea.config.path, idea.system.path, idea.plugins.path and idea.log.path options
Thank you Elena, it looks like that should work. Good detective work on finding that link; I couldn't find the right thing to Google on. Seems like it might be a JVM thing. Weird.
Thanks for your help.