App needs elevated privileges to run - why?

As per one of the comments, your best bet is likely to be using Sysinternals' Process Monitor to see what is being tried and failing due UAC.

Download Process Monitor from http://technet.microsoft.com/en-us/sysinternals/bb896645.aspx

Run process monitor, run your borked, obsolete, badly coded app, go to process monitor > filter > if Process Name is <borkedapp> then Include and if Result is Success then Exclude which should give you a mere few thousand entries to scan for the bit that is failing.

Alternatively there is a tool linked below (I have not tried it) that will help you compare process monitor traces so you can take a trace running as standard user and another running elevated and compare them.

http://forum.sysinternals.com/tool-for-comparing-procmon-traces_topic28870.html

https://github.com/patraulea/LogDiff

tIn addition to Blue's answer, there are tools for migrating applications to Windows 7. Even though your app says it's Windows 7 compatible, running it through this ringer would tell you what is needed for your app to correctly function with UAC on. You may even be able to create a 'shim' which modifies certain settings when your app runs. I believe the tool is called the Standard User Analyzer http://technet.microsoft.com/en-us/library/cc766021(v=ws.10).aspx.

I'll also add that a lot of the times the elevated privileges can be avoided by granting the appropriate permissions, i.e., giving the user modify rights to the C:\Program Files\ folder and subfolders.