Let's get to the core question:
My manager is asking me if a Windows 32 VB6 app will work on 64 bit Windows 7.
The answer is "probably, but you can't be sure and you might have to make a few tweaks to your system". Most apps that ran on XP-32bit will run on 7-64bit with no extra work. There are rare exceptions where the 64bit instruction set causes problems.
Much more common are issues where rules from Windows XP are now more strictly enforced in 7. For example, in Windows XP, standard users do not have write access to the program files folders, but so many users ran with administrator privileges a lot of developers got away with one here and never even knew it. Windows 7 now does a better job of enforcing the same rule that always existed, but in so doing it can break some old apps. Usually you can get around this particular rule by manually giving write permissions to the Users group for just the specific folder you need, but this is just one example.
But the main thing here is that it's unlikely to be the 64bit vs 32bit issue that's causing you problems.