Is it possible to get my headset to initiate a recording app and record my voice? I have a set of Sony headphones with a single button and microphone. I also have a Samsung SC-04F running Android 5.0, unrooted.

I want to be able to press the button on my headset once and record my voice. When I press the button again, the recording stops.

Ideally this can happen regardless of what ever other app I happen to be using. So, if I'm listening to music, pressing the button will start the recording app. I guess it should probably pause the music, and then ideally return to music playing when I stop.

I've looked at options for apps that claim to allow recording via the headset microphone. But, this one for example, seems to require that the app already be running. I've also looked at headset button control apps, and I can get it to open a voice recorder app, but not actually start recording.

Is what I'm trying to achieve possible?
I can't say too much here, but this thing may be achieved by an app called Tasker. Many users say it god of android. To function on every aspect of requirement, it needs root access. In your case, it does not require root access.

You can use its capability to achieve that certainly. This is wiki page of that app.

Edit :
As OP said that he has hands on Headset Button Controller. This app is also supports tasks created by Tasker app. 
Steps to Achieve this ->


  
  Open Tasker App. Go to Tasks tab. Click Plus(+) to add a task Record. Now edit the task to get it function what you want. 
  Click +, then click on Media , then Record Audio.
  Give the file name of your recorded file. Select your source ( In this case Microphone). You can select format on your choice.
  Now link this task with that app. Open app. Click on one of interaction that you want (Single click is your choice in question). 
  Now choose Tasker task and select the task (Record) as you created in Tasker app.
  You are good to go. Try it.
  


I tested it by myself. So, it must work. On more out of Tasker, use WiKi as posted above.