Where does it say 'You Are Already Enlightened'?

It is not zen, but the famous puthujjana called Third Karmapa, Rangjung Dorje has a short exposition on the doctrine that they created http://www.rinpoche.com/teachings/buddhanature.htm

   “All beings are Buddhas,
         But obscured by incidental stains.
         When those have been removed, there is Buddhahood.”
         That is a quotation from a Tantra.

he says that the characteristic of his doctrine is more about the nature of his attainment

The mind that has the absence of the three obscurations
Is “(the wisdom of) equality” and it is “peace.”
Due to having love and great compassion (for beings)
The sambhoga(kaya), etc., appears to them.
This is stated in order to refute those who say
That the attainment of Buddhahood is the same as the Hinayana (attainment).

Wisdom is the three permanences:
Permanence of nature is the dharmakaya;
Permanence of continuity is the sambhogakaya;
Uninterruptedness is the nirmanakaya.

There are three impermanences:
Mentally fabricated emptiness is impermanent;
The mind of moving thoughts is impermanent;
The composite six consciousnesses are impermanent.

However, the three permanences are present.
The three impermanences are stains.
The three permanences are wisdom.

This is not the same as the Tirthika “self,”
Because that is a mental fabrication and (Buddha nature) is not.
This is not the same as the nirvana of the Shravakas and Pratyekabuddhas
Because (in that) all the qualities of the form kayas are not manifested.
This is not the same as the body of an (ordinary) being
Because it is not created due to the defilements.

So I heard,

Don't let anyone tell you that you're not enlightened.

Not formal Zen lineage, but as Andrei wrote, "the phrases are more nuanced and the meaning is more subtle".

As in Zen, Vajrayana too, needs to “assume” that oneself is already enlightened. That assumption is removed by having a vajra guru, through a process known as “initiation”.

Since only an enlightened being can see another enlightened being, only if oneself is already enlightened that he can realise his own enlightenment. That is the starting point. That was precisely how the Buddha did realised his own enlightenment, he was able to fight off the last 3 illusions under the bodhi tree.

From there he was able to teach the entire path to enlightenment starting with Theravada.

It's the most standard Mahayana teaching. The references are too numerous to cite, almost every other text has statements to this effect. However, it should never say "You are already Enlightened" (or anything naive like that), the phrases are more nuanced and the meaning is more subtle.

For example, it may say, "Our nature is fundamentally pure" or "Your original nature is no different from that of the Buddhas" etc. - in an attempt to turn our attention to the existential conflict between our dualistic mind of "this is wrong and I need to be something else" and the perfect suchness of Nirvana in the here-and-now.

Now, if you speak with an actual Zen (or another Mahayana) teacher, they will be quick to point out that even though our primordial nature is indeed Nirvana, the habits of craving and clinging are extremely difficult to overcome on the spot, which is why we must cultivate Sila/Prajna/Samadhi.

So in some sense the teaching has two sides and which one your teacher will drum depends on which way you lean in your particular state of confusion. If you are inclined to be complacent they will drum the Path and if you are obsessed with the goal, they will drum Buddha-Nature. It's kinda frustrating because you end up being wrong no matter which position you take. It's only when you mature beyond positions is when this contradiction resolves.