It is DM fiat.
There’s nothing explicitly preventing this, but it would be funny to have that NPC be charmed by the familiar and not by the warlock speaking throught it.
The effects of the charmed condition are two:
- A charmed creature can’t attack the charmer or target the charmer with harmful abilities or magical effects.
- The charmer has advantage on any ability check to interact socially with the creature.
I would say as a DM you can make this work, but remember that the charmed creature is in complete control of their own actions and would still behave as it would have had if they weren’t charmed: they simply are more susceptible to what the charmer might say. For instance, even if you charmed the king, said king yould never forgo their position just because you are charming it.
In conclusion: the charmed condition is common through many spells that manipulate the target, but this feat does not include the extra benefits of those spells: it’s just applying the charmed condition.
That, and I still think it would be hilarious to have the familiar charm another person this way.