In Maya, you can do FK/IK matching with Python by getting the current transformations of the controls and doing some calculations. Is there a way to use Python to manipulate controls in Apex when animating(apex scene animate)? At which step should FK/IK matching be implemented (where should the scripts live and how do the user activate the script)?

I found a pretty dumb solution for matching from IK to FK. I created some dummy controls that track the correct position for FK controls by getting the transform of the deformed skeleton. Then when I need to transition from IK to FK in Apex scene animate, I copy the transform of the dummy controls back to the corresponding actual FK controls. Then I change the mode from IK to FK and I can keep animating in FK.

The same should work for matching from FK to IK.

Is there a way to automate this process?

In 20.5, unfortunately no. For the next major release, we've added the ability to attach python script events to controllers which will allow one to easily build a rig to do this.