Bill,
Use the shutdown -Fr command. This shuts the system down immediately (that's what the -F is for) and brings the system back up after it is shutdown (that's the -r). On some Unix platforms, there is a subtle difference between shutdown and reboot. I don't think that is the case with AIX, but shutdown -Fr will accomplish the same thing as reboot.
I can't see what the difference is between shutting down followed by bringing the system back up again immediately and shutting down and turning it off. (On some systems, you have to literally unplug the thing to cut the power.) The system's memory isn't going to know if you unplugged it or not. Once it's shut down, the memory is cleared. Maybe someone else knows something you and I don't, however.
Yes, these guys should go back to school to fix their memory leak and learn more about AIX. I had a similar experience a few years ago when the vendor told us we had to boot the system every week, but, of course, they didn't say it was to fix a memory leak (but everyone knew that's where they were coming from). My boss had a background with hospitals and pointed out that those hospital systems couldn't be booted for medical reasons so he couldn't understand the reason for weekly boots. Our Unix sys admin guru told the head programmer (in a rather brutal fashion, but it worked) to fix his program because the most we would boot the box would be monthly (and then only reluctantly). The vendor fixed the program.
I'd ask your vendor what Unix platform their app was designed on, too. If it was programmed on some obscure Unix platform and then ported to AIX, their porting it to AIX might have introduced some memory leaks and they don't know how to fix it on the AIX platform. Ask them how much AIX experience they have had and how many AIX installations they have put in.
I think you should get a new vendor!
Good luck. Let us know how it turns out!