Your familiar can deliver the spell as if it had cast the spell.
Inflict Wounds includes a melee spell attack. The rules for spell attacks state (PHB p. 206) that:
Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
The Elven Accuracy feat (XGtE, p. 74) states that
Whenever you have advantage on an attack roll using Dexterity, Intelligence, Wisdom, or Charisma, you can reroll one of the dice once.
The second person pronoun is used in both of these last two wordings. So if the caster's, rather than the familiar's, spellcasting ability modifier and proficiency bonus are used for this melee spell attack, it seems to follow that, given advantage on the attack, a reroll would be available via Elven Accuracy (if the caster has that feat). Is this the case?