Find Familar spell, Familiars can't normally attack.

But

Pact of the Chain can summon imps and pseudodragons that have attack actions baked into their statblock.

In addition with Investment of Chain Master

I can forgo an attack to use my reaction to call the imp to attack 

Or use my Bonus Action to cause it to attack.

But on it's turn can it attack normally using its action to attack, like it says in the statblock, or not?

No, I don't think it can.  I think out of the box, it dodges unless you use one of your own abilities to make it attack.

Almost every creature has an attack baked into its statblock. A poisonous snake has an attack baked in too, but when you have your familiar as one of them it can’t attack. Neither can that Imp.

The stat block is not just for the find familiar spell it is for encountering the monster normally. This is the case for all other familiars as well. When using the find familiar spell, whether that is through chain or otherwise follow the rules of the spell. In this case the Imp is not fully an imp it is a familiar with the imp stat-block and must follow all the rules listed under the find familiar spell as with all other familiars. The conditions in which it can attack are given via the pact of the chain text.

RAW (2014 and Tashas), when the warlock makes the Attack Action, you can forego one of your Warlock attacks, (most likely the only one you have), to allow your familiar to attack with its Reaction on your turn.

In addition, Investment of the Chain Master allows you to give an command to the familiar as a Bonus Action that allows it to take the Attack Action on its turn.

So the imp can attack once on your turn, and once on its turn, BUT that will cost your Action (you have to take the attack action and forego it) and your Bonus action. That means, the Warlock will not cast spells or even cantrips.   

Something else to remember, and it is always frustrating because it feels so darn pedantic to say, but is important when summoning things. You are not conjuring the actual creature, you are summoning a spirit that is assuming the physical characteristics of the creature. The spirit is beholden to your whims and awaits your commands to act unlike a creature with free will. The summoned spirit will defend itself, but that is the extent of its ability to act outside of the caster's commands.

If you befriend a creature and it willingly commits to being your familiar, then you can bind it to you in ceremony and it will act freely of your commands. This also can create a predicament where the creature decides to not listen to you, disappear at times, switch loyalties, or even work as an adversary when displeased with its treatment. Think of trying to wrangle an imp or quasit into doing good deeds without some ulterior motive to gain from it. It basically introduces a character the DM can assume control of if the story calls for it.