I've seen the clarification in the PHB Errata which states:
Pact of the Chain (p. 107). When you let your familiar attack, it does so with its reaction.
However, I wondered if there had been some sort of further official clarification on this since the errata came out… It also raises the question as to whether the familiar would still have both its movement and action left for the turn in which it attacks.
Best Answer
It's a reaction, as stated in the errata
The errata clearly states that when the Warlock lets their familiar attack, the familiar uses their reaction to do so.
No, because the familiar didn't necessarily have actions or movement on that turn in the first place - the Pact of the Chain familiar can only attack when the warlock takes the Attack action, which usually happens only on the warlock's turn. It can also happen on other turns, even the familiar's own, if the warlock Readied an Attack action, but no matter the turn the Attack happens on, it's always a reaction for the familiar to make the Pact of the Chain attack. If it happens on the familiar's own turn, it doesn't lose any movement or actions for it.
The familiar will have actions and movement as usual on their own turn, but cannot use their action to attack on their own because of the limitations of Find familiar (spell description on D&D Beyond).
Pact of the Chain exception to the above