Landing a killing blow in V.A.T.S. will immediately restore all Action Points, allowing the beginning of a new series of attacks immediately. However, foes killed by allies (or other means) will not trigger the perk, even while in V.A.T.S.
Fallout: New Vegas
In Fallout: New Vegas, the perk only gives back 20 AP on a kill in V.A.T.S. Although this is a significant reduction to the effectiveness of the perk, the perk is still very useful with high power, low AP cost weapons. The effectiveness of Grim Reaper's Sprint is dependent upon three things: AP cost of the weapon, damage of the weapon, and toughness of the enemy.
Be aware that the perk restores 20 Action Points if an enemy is killed in V.A.T.S., not per enemy killed; e.g. killing 3 weakened enemies in a single round of V.A.T.S. will only restore 20 AP, not 60.
Any kill in V.A.T.S. has a 35% chance to restore all Action Points and refill your Critical meter.
When fully maxed out, this perk can be incredibly useful in the mid to late game, particularly in large battles. This is because higher level enemies are often accompanied by lower level "cannon fodder" types, who can be instantly killed at higher levels. These enemies essentially provide a reasonably reliable source of regaining Action Points, since at higher levels they can usually be killed in one or two shots, with a 35% (at max level) chance of activating Grim Reaper's Sprint.
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Knocking an essential non-player character unconscious counts as "killing" them for the purposes of this perk.
In Fallout 3, the G.E.C.K. script actually gives 1000 AP after a kill, rather than restoring the exact amount of AP.
Be aware that in Fallout: New Vegas, some weapons will not gain another shot in V.A.T.S. from only 20 AP. Even in this case, the perk retains some usefulness in that it does allow going back into V.A.T.S. sooner and fire again, even if one cannot immediately return to V.A.T.S. and fire again.
In Fallout 4, Kellogg's pistol or any weapon with the Relentless effect can serve as a means of saving three perk points with 100% chance of refill on crit instead of kill as with this perk. Pistols with reflex sight work especially well, since they require a tiny amount of AP to fire. In addition, Kellogg's pistol when fully upgraded has a crit damage that is quite hefty even without Gunslinger.
In Fallout 76, unlike in Fallout 4, there is no icon or sound effect when the perk procs, making it difficult to tell if it is functioning properly.
Behind the scenes
Grim Reaper's Sprint was a fan-created perk and winner of the Fallout 10th anniversary perk contest in 2007, submitted by Marc-Andre Deslongchamps. Bethesda implemented the perk in Fallout 3, describing the idea as "elegant and simple" while affecting game mechanics in a unique way.
Sometimes upon taking this perk, it will not restore AP. Instead, it just takes up a perk slot that could have otherwise been used on something else. [verified]
APs gained through gear in Fallout 4 will not restore. (a total of 195 APs, 30 APs through ultralight mod, Grim Reaper fills APs up to 165.) [verified]