I investigated, and Google Groups helped me find the solution in a microsoft.public.win32.programmer.ole thread. The problem was the .dll file I had built was built in Visual C++ 6 in "debug" mode .dll; the "debug".dll (unbeknownst to me at thetime) had extra .dll dependencies which were present on my machine, but not on the tester's machine.
The solution was to build the .dll in "release" mode instead.
Google also turned up a helpful cryer.co.uk article which lists (1) "The name of the COM object is wrong" and (2) "The COM object requires another library which is missing" as possible causes of the "The specified module could not be found" regsvr32 error. The article suggests using the Sysinternals Filemon utility to determine which .dll file(s) are missing.