Does the Chosen Disciple perk work outside of VATS?

Solution 1:

Although the wikia is not terribly clear on this:

Kills made by your melee weapons will restore some of your Action Points.

This guide says that you will get AP for melee kills always with this perk.

The Disciples' bonus perk is very beneficial for melee builds that utilize VATS. Any kills you make with a melee weapon will restore 'some' of your Action Points. From peeking at the game code, it would seem this gives you 25% of AP back on kill 100% of the time.

I think the reason they mention VATS in this post is because you would be able to use VATS much more frequently with the 25% AP regeneration effect.

Solution 2:

Yes, you will regenerate AP outside of VATS if you have this perk. From the Wiki:

Kills made by your melee weapons will restore some of your Action Points.

It does not specify whether or not you need to be in VATS, but its implied that you do not.