No, VT only allows you to "make the attack again on each of your turns as an action." So, two restrictions: it has to be your turn, and you have to use an action.
The only way to use it as reaction with War Caster is if you cast it for the first time (or use up another spell slot) when an enemy provokes an OA.
Spells that allow extra attacks on subsequent turns specify the action you can use to repeat them: look at Storm Sphere, Investiture of Flame (which allow repeating an attack using your Bonus Action), and compare them to Vampiric Touch (or Witch Bolt).
Your GM can always allow it, but RAW, it doesn't work.
The only way to use it as reaction with War Caster is if you cast it for the first time (or use up another spell slot) when an enemy provokes an OA.
Spells that allow extra attacks on subsequent turns specify the action you can use to repeat them: look at Storm Sphere, Investiture of Flame (which allow repeating an attack using your Bonus Action), and compare them to Vampiric Touch (or Witch Bolt).
Your GM can always allow it, but RAW, it doesn't work.