If a Warlock's familiar (via Pact of the Chain) delivers a touch spell, who provokes opportunity attacks — the familiar or the warlock?

Neither!

In D&D 5e, simply casting or delivering a spell does not provoke an Opportunity Attack.

Here is how the mechanics break down:

• The Warlock: Casting a spell while in melee no longer provokes Opportunity Attacks in 5th Edition.
• The Familiar: When you cast a spell with a range of touch, your familiar can use its Reaction to deliver the touch for you D&D 5e SRD 5.2 — Spe.... Making this touch delivery also does not provoke an Opportunity Attack.

The Catch (Movement): Opportunity Attacks in D&D are triggered entirely by movement — specifically, when a creature that an enemy can see leaves their reach D&D 5e SRD 5.2 — Gam....

So, if your familiar flies up to an enemy, uses its Reaction to deliver your Shocking Grasp or Inflict Wounds, and then moves away on its own turn, that movement is what would provoke an Opportunity Attack against the familiar D&D 5e SRD 5.2 — Gam....

Pro-tip for Chain Warlocks: Opportunity Attacks require the attacker to be able to see the creature leaving its reach D&D 5e SRD 5.2 — Gam.... Since Pact of the Chain special familiars like the Imp or Sprite can turn invisible, they can easily slip away after delivering a spell without provoking any attacks!

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