A direct reading of the rules does not allow Vampiric Touch to affect the mount provided by Find Steed.
The requirement of Find Steed is that the spell targets only the caster:
While mounted on your steed, you can make any spell you cast that targets only you also target your steed.
The text of Vampiric Touch specifies that there may be targets other than the caster (emphasis mine):
On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt.
Hence, such spell does not fulfill the main requirement for having the mount affected by the spell.
Moreover, recall that the "range" entry in the spells description does not correspond to the target(s) of the spell.
A DM can rule otherwise.
There is still a debate about what is a target for a spell, since the rules are not completely consistent. See for example the rules for Range:
The target of a spell must be within the spell’s range. For a spell like magic missile, the target is a creature. For a spell like fireball, the target is the point in space where the ball of fire erupts.
and the description of Fireball (see the discussion in this answer about this topic).
For spells with range of Self, the rules say (emphasis mine):
Most spells have ranges expressed in feet. Some spells can target only a creature (including you) that you touch. Other spells, such as the shield spell, affect only you. These spells have a range of self.
Under this description, it seems the spell with range of self effectively target the caster. But Vampiric Touch specifies (emphasis mine):
Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt.
A loose interpretation of the description may suggest that the actual target of the spell is the caster, while "target" in spell's text refers to the target of the particular attacks granted by the spell and not to the spell's target.
The next section covers this last interpretation.
Under the magic items rules, only the proficiency bonus of the mount applies for its attacks.
If the DM allows Vampiric Touch to affect also the mount, the bonus for the attacks is given by just the proficiency bonus of the mount, since it has not any spellcasting ability. This can be inferred by the rules for Magic Items in the DMG, page 141:
If you don't have a spellcasting ability-perhaps you're a rogue with the Use Magic Device feature- your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.
Indeed, who is making the attack is the mount, not you (the original caster), so it has to use its own abilities. Compare with the Faithful Hound spell, which on the other hand specifies
The hound's attack bonus is equal to your spellcasting ability modifier + your proficiency bonus.
See also the table in the Sage Advice Compendium about How do I know which ability modifier to use with an attack roll and its damage roll? .