I'm having considerable difficulty running a 16 bit .exe. The error message says I don't have the permissions. One link said that I should configure the computer in group policy, but I'm not sure how to do that. I'm running Win 7 Ultimate x32.
If you are sure that it is a 32 Bit version of Windows 7, then it should still support 16 Bit Apps as that was built into the architecture and not the OS itself. Much like the RAM support.
If you have the 32 version in theory you should be able to run it, you may need to set the compatibility to be something like windows 95 or windows 98/me
but if you actually have 64bit then I think something such as DOSbox is the best way to run it
Since you're running Win 7 Ultimate you can download and install XP Mode from Microsoft which is a Virtual PC environment that includes a fully licensed version of Windows XP. Using XP Mode will allow you to run that app from your Win 7 desktop.