Yes, for most available options
PHB 240, emphasis mine:
...when you cast a spell of range touch, your familiar can deliver the spell as if it had cast the spell.
Because each metamagic option begins with "When you cast a spell" or "When you roll damage for a spell", the familiar can deliver your metamagicked spells, because the familiar is not casting the spell, it is delivering the spell you cast as if it had cast it.
This does not work with Distant Spell. Distant spell changes the range from touch to have some range (see this question), so the familiar cannot cast a touch spell that has been made distant.
Note: If a metamagic option is ever released that uses different language, this answer may not apply because instead of the language "When you cast a spell" being in the Metamagic feature, it is in each of the Metamagic options. Language that may change this answer could be: "When you touch a creature while casting a spell with range touch" or something similar.