First of all, thank you so much everyone (especially MarteenDee), AP is much more fun with all those customizations proposed here!
Now, I've experimented a bit with <AP Program Folder>\APGame\Config\DefaultInput.ini, and it turns out reducing the "Speed" values in the movement bindings in [APGame.APPlayerInput] to something between 0.2 and 0.4 actually leads to a walking animation.
So by simply copying the four movement lines and replacing the binding and the speed values, you will get to choose if you want to run or walk at any moment. I mean something like this:
+Bindings=(Name="MoveForward",Command="Axis aBaseY Speed=1.0")
+Bindings=(Name="MoveBackward",Command="Axis aBaseY Speed=-1.0")
+Bindings=(Name="StrafeLeft",Command="Axis aStrafe Speed=-1.0")
+Bindings=(Name="StrafeRight",Command="Axis aStrafe Speed=+1.0")
...
+Bindings=(Name="W",Command="MoveForward")
+Bindings=(Name="S",Command="MoveBackward")
+Bindings=(Name="A",Command="StrafeLeft")
+Bindings=(Name="D",Command="StrafeRight")
+Bindings=(Name="<Key1>",Command="Axis aBaseY Speed=0.2")
+Bindings=(Name="<Key2>",Command="Axis aBaseY Speed=-0.2")
+Bindings=(Name="<Key3>",Command="Axis aStrafe Speed=-0.2")
+Bindings=(Name="<Key4>",Command="Axis aStrafe Speed=+0.2")
I have been experimenting a bit with the ",Control=False,Shift=False,Alt=False" modifiers, too, hoping to be somehow able to use one of those keys as a walk modifier, but to no avail: I haven't been able to bind two different commands to the same key (with different modifiers).
Also note that changing the APInput.ini in "My Documents" alone doesn't seem to suffice; you need to edit the DefaultInput.ini.
BTW, walking while crouched actually looks like sneaking!