No. The Bonus Action attack you make with two weapon fighting is not the Attack Action. To use this feature, you must replace one of the attacks (lower-case a) granted to you by the Attack (upper-case A) Action. And nless your warlock is multiclassed, the Attack Action should only grant one attack.
You also can’t replace the attack granted by the Attack Action and still make a Bonus Action attack with your off-hand weapon.
In this case, you are taking the Attack Action and your familiar is attacking. You are not taking the Attack Action and making an attack with a light weapon that you’re holding in one hand. However, if you have the Extra Attack feature (say, from taking 5 levels in Fighter), then you get to make two attacks when you take the Attack Action, so you can replace one of those attacks with an attack from your familiar and use the other one to attack with a light melee weapon you are holding in one hand. So in that case, you could then use a Bonus Action to make an Attack with a second light melee weapon you are holding in your other hand.
Of course, all of this is a very strict RAW interpretation, and it’s possible your DM will rule differently.