No, War Caster doesn't allow Vampiric Touch as an Opportunity Attack
Vampiric touch says (PHB, pg. 285):
Casting Time: 1 action
Make a melee spell attack against a creature within your reach.
Until the spell ends, you can make the attack again on each of your turns as an action.
The War Caster feat says (PHB, pg. 170):
When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.
So War Caster needs the spell to have a casting time of 1 action (check) and needs to only attack one creature, but the spell has a range of self, so what is range?
The Range section says (PHB, pg. 202):
Most spells have a range expressed in feet. Some spells can target only one creature (including you) that you touch. Other spells, such as the shield spell, affect only you. These spells have a range of self.
This seems to split Touch range and Self range spells, and specifies that, although vampiric touch has you make a melee spell attack against a creature, the spell itself only affects you. Therefore, due to the range of the spell, you could not cast vampiric touch for your opportunity attack via War Caster, because War Caster needs a spell that targets a creature, not yourself.
However, once cast, you can use Vampiric Touch for Opportunity Attacks
That's when the spell is first cast. What about if the spell is still active from before (assuming you are maintaining concentration)?
The Opportunity Attacks section says (PHB, pg. 195):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
Since vampiric touch only requires you to make a melee attack, this should be fine, since the opportunity attack rules do not specify melee weapon attack.
However, the last part of the quote I've included for vampiric touch says "as an action". But it seems to me that the rules for an opportunity attack are more specific than those of normal combat actions, since you're being allowed to make attacks outside of your turn, whereas the description of vampiric touch is describing normal "on your turn" combat.
Hence I conclude that you can keep using vampiric touch for opportunity attacks for as long as the spell is active.