[RPG] Can a Druid/Warlock multiclass use Voice of the Chain Master to speak through their familiar while in Wild Shape

No.

Consider that:

Verbal (V)

...the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion.

If you can argue that this is ALL that is needed to cast a purely verbal spell, then anyone or anything with a mouth can cast it.

Let's get more basic: suppose you don't have spell slots left to cast Healing Word. Will speaking the words that produce "the particular combination of sounds, with specific pitch and resonance" set the magic in motion?

It all boils down to the power to cast it, and I'm afraid your familiar does not have it.

Let's look more closely, then, at:

Voice of the Chain Master

Prerequisite: Pact of the Chain feature

You can communicate telepathically with your familiar and perceive through your familiar’s senses as long as you are on the same plane of existence. Additionally, while perceiving through your familiar’s senses, you can also speak through your familiar in your own voice, even if your familiar is normally incapable of speech.

There is nothing in there that ever mentioned -even vaguely- casting of any sort.

The argument of specific-beats-general here is also invalid because "specific" also denotes "explicit" and there is nothing here that explicitly and specifically grants you the power to cast verbal spells through the familiar. This case is more of a general-beats-vague.

Finally, the Find Familiar spell states the times when a familiar can (sort of) cast a spell:

Finally, when you Cast a Spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

Voice of the Chain Master lacks similar text that lets you cast verbal spells through it, so the feature can't let you do that.

There likely isn't a RAW answer, but...

Being permanently mute is not a condition that the core rules represent, so any answer here is going to be speculative and highly subject to the DM's rule on the matter.

However, I would encourage a DM ruling on this to allow it, particularly because the character is taking on a significant disadvantage that isn't compensated for anywhere else in the rules. So giving a free "out" isn't exactly letting the character get away with anything.

A more important question for you before you decide to volunteer your character for this disability is how you plan to use spells that have a verbal component. A character who can't speak (due to a gag or silence spell) generally can't use verbal spells, and speaking through your familiar may or may not make up for this limitation, according to your DM.