After unsuccessful web and forum searches and a ton of experimentation, I finally figured out what was going on. For whatever reason, Traktor tries to access your installed storage devices when it starts up. My computer doesn't have a floppy disk drive (not may do any more), but Windows had a "Drive A:" device registered anyway, and whenever Traktor tried to access this non-existent device, the error would occur.
Here's how I fixed it: I opened up the Windows Device Manager, found the bogus drive entry, right-clicked it, and disabled it.
Windows warned me that it would cause the device to no longer function (no big deal since there was no device in the first place), and after that the drive appeared disabled in Device Manager.
Now Traktor no longer pops up the error, and it actually starts a bit quicker too!