There's no reason you shouldn't be able to do this. There are licensing issues, however.
The software does not store anything in the Windows registry (except for the license). So it takes the folder from which it is run as its "home" folder. Program components must be in the correct places
in that folder. This makes it essentially portable.
The location of the preferences file, the data store file, and the default Data Store will be determined by the system, since they are in the currently logged-in user's Documents folder. That presents a slight issue, since it will
always look in that folder. I'm not sure how one would handle this today.
In version 3.2 (still in beta), you can create a file in the same folder as the TD which will specify where to find the preferences and data store files. This would allow you to store the preferences file, the data store file, and the Data Store itself on the flash drive. The only issue I see here is that the flash drive might be mapped to a different drive letter on different PCs. If you can map it to a consistent drive letter, it should work.
But your TD license won't travel with you, since it is stored in the registry. Meaning you'd have to use the trial period, or install your license on any machine you use. Here's the FAQ on that:
http://www.thetournamentdirector.net/faq.html#faq3