The answer is that there really isn't a standalone version. When you install VS Team Explorer on a machine without Visual Studio, the installer will install a Visual Studio shell. Then, when you run Team Explorer in standalone mode, you are actually running a Visual Studio shell.
Martin had a good point about Team System Web Access, which probably would do the job nicely. Plus it has the added benefit that it allows non Visual Studio users access to work items. But, it was decided that it was too much trouble to get permission to install it here (working for the US Army can have its issues).
My solution for now is to run another instance of Visual Studio and access the Team Explorer tools from there.