I am playing a warlock. I will take the Pact of the Chain option at 3rd level to have the Find Familiar spell.
If I cast invisibility on myself and then on my turn always take the Attack action to allow my familiar to attack, do I stay invisible?
Strictly speaking, yes.
The spell ends for a target that attacks or casts a spell.
It doesn't say that it ends for a target that takes the Attack action, so if you take the Attack action and don't attack, then RAW the invisibility stands.
Additionally, when you take the Attack action, you can forgo one of your own attacks to allow your familiar to make one attack of its own.
Taking the Attack action and then having your familiar do the attack is not you attacking.
You can refer to this post for further clarification on the differences between the Attack action and making an attack: What does upper-case-A-Attack action vs. lower-case-a-attack mean?