[RPG] Can a mute warlock make normal use of the Voice of the Chain Master class feature

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.

First, we will assume that your DM has deemed that the Telepathy of the Warlock is 2-way, as it was intended as only one way. You don't technically need the Great Old One's ability as Voice of the Chain Master allows this anyway.

Second, if you attack an opponent you can't see you would still have disadvantage, the only gain you would get with this combination is that you can help them target the correct location on the field of battle as per PHB page 194 and 195.

Also, you can do the same thing with Devil's Sight invocation instead and cut out the "Middle-Imp". It seems that you have a lot of redundancies and a DM, such as myself, could deem that you would be unable to point out but one or two of the occupied locations but not more as it would take time for the imp to communicate to you then you to your colleagues. This of course would have to be done using no visual queues which a kind DM would overlook but another could easily require int checks to help in targeting while interpreting the Warlock's instruction.

Bottom line is that your teammates still cannot see them so they still have disadvantage, they just won't have that auto-miss chance for targeting incorrectly. This all assumes the targets don't take the Hide action. There is a difference between unseen and hidden in the rules.

Additional note, the Devil's Sight that is granted by the invocation is better than the Devil's Sight that is present on devils, as the invocation allows you to see normally and the devils simply use their darkvision within it, which means they have disadvantage on perception based on sight.

You can see normally in darkness, both magical and nonmagical, to a distance o f 120 feet. PHB page 110

Versus

Devil's Sight. Magical darkness doesn't impede the devil's darkvision. MM page 70