Once I have cast vampiric touch I can make a melee spell attack "on each of [my] turns as an action".
In the first round, the round I cast the spell my action is to 'Cast a Spell', and I clearly can not use Two-Weapon Fighting. However, in subsequent rounds my action is to make an attack. Does this attack allow me to use the Two-Weapon Fighting rule and make an attack with a light weapon in my off hand as a bonus action?
Two-weapon fighting stipulates that you must make an attack with a weapon. So the crux of the question:
Is a melee spell attack with vampiric touch equivalent to attacking with a weapon for the purposes of using Two-Weapon Fighting?
I would consider a hand that can do 3d6 necrotic damage a weapon.