Both require an action. For shocking grasp to deal damage, you need to cast it using the Cast a Spell action. Its casting time is 1 Action.
The description of vampiric touch states:
Until the spell ends, you can make the attack again on each of your turns as an action.
You only have one action, thus only one of them can be cast per turn. You might be confused thinking that vampiric touch replaces your melee attack or that when you attack its effect would activate - but that's not the case. You have to use your action to specifically attack with vampiric touch.
One exception would be using Action Surge as a multiclass Fighter/Wizard and casting both spells using two separate actions. That's probably not what you intended, though, since you ask on the same attack roll as well (and that would be impossible since these are two different actions making different attacks).