I have inherited a Flex application which needs some minor changes.
What software do I need to install to be able to build the application?
I have written Delphi/C++/SQL code for nearly 30 years, but have no experience with Flex. I maintain the database for the system, but the UI developer died unexpectedly.
It depends on how the UI developer had it set up and the details of the application. Your best bet is to get access to their machine and examine the workspace they have set up.
At the very least, you need to install the relevant Flex SDK from Adobe or Apache (Adobe donated the Flex SDK to Apache in 2011, but I don't know whether a legacy application would use the Apache version).
You'll also need an IDE -- Adobe's Flash Builder, FlashDevelop, or FDT (search Flex FDT) are the most likely tools to download.
You may also need Ant or Maven.
The simplest path for Flex apps is to write them in Flash Builder and hit the Publish button to deploy, but scripted builds for more advanced applications were also fairly common with Flex.
To find out which version of Flex you need to use is a little harder. Unfortunately, it's not guaranteed to be listed in any one location. These questions may help, but it's definitely a trial-and-error process:
How to know exact flex sdk version from existing project files?
Is older version of flex SDK really required?
Flex Builder cannot locate the required version of flash player