What platform are you on? It may not be possible, but there are a few options you can investigate depending on the OS you are using.
See if the other application is AppleScriptable. If so, you may be able to set up FileMaker to run an Applescript that extracts the needed data from the other app and sets a field or fields in your database to that value. I haven't touched Applescript in years, but my memory is that such should be possible but only if the other application was designed to permit the interaction with AppleScript.
I will investigate. I also wonder if support for Assistive Devices could be used?
The app does not use AS. There must be another way.
After some research, I believe this is done via AppleScript's GUI Scripting (Enable Assistive Devices) to control the UI of the application I need, copy text and paste into specific FMP fields. DOes anybody have any experience with this?
It depends on the application you are talking about. Some apps have applescript libraries (the things Applescript can access) some don;t.
There are sites with AppleScript information and the Apple Discussions are good source.
Introduction to Scripting FileMaker Pro
AppleScript and FileMaker