I have a program built with VB6 and using some 32-bit DLL\'s. Will this program run on a 64-bit machine? This page suggests that it should run fine on the Windows on Windows
Yes, both the VB 6 program and the DLLs will work just fine on a 64-bit version of Windows.
Since both are 32-bit, they will run under the Windows-on-Windows 64 (WoW64) subsystem, which effectively emulates a 32-bit operating system on the 64-bit versions of Windows.
I've run many such programs myself with nary a hitch.
Beyond Explorer extensions and kernel mode drivers (neither of which you've written in VB 6), any compatibility problems that you might experience are almost certainly the result of bugs in your own code, which are easily fixed upon detection. Ask more questions about that when you find them.