Isn't at least one of the Zen traditions - Rinzai Zen - Vajrayana?
There are indeed a number of branches of Zen, all of which follow Chinese traditions, and all of them are very much Mahayana in view and method, Rinzai Zen included.
Zen teachings sound quite close to some of Vajarayana teachings
One interesting aspect of Zen which it does have in common with Kagyu Dharma is that it is based on the third turning of the wheel of Dharma teachings, ie, on those of Tathagatagarbha (Buddha Nature) teachings, and on the 'Three Natures' teaching. To that extend, it also has a flavour to it which very much focusses on the fullness of
qualities which are present in the Enlightened mind, and present in Emptiness, which Kagyu Dharma also has.
Another commonality with Kagyu Mahamudra is that Zen practice, like that of the Theravada, is very simple in its core and that simplicity is in many ways its strength, for those that are karmically attracted to that path. Whether you are doing Koan practice in Rinzai Zen, or doing 'Just sitting' practice in Soto Zen, either way, there is very little for you to work with, very little for you to get caught up in and lost in (such as intellectual ramifications), and so you are forced to focus your efforts Dharmically. For those so inclined, this can be a very powerful means indeed. Sometimes simple is indeed best!
However, as noted above, the practice is very much on the causes rather than on the results, so is not Vajrayana. Of course, that does not diminish it's effectiveness or value, if that is what you are karmically predisposed to. And of course, much of Kagyu Dharma is not Vajrayana either, for that matter.
For Lin Zi personally, I have very special gratitude.
thank you for sharing your very inspiring description of your inspiration from reading and reflecting on Lin Chi. Many of the Chinese and Japanese hermits writings are so stunningly direct, and evocative indeed.
with best wishes in the Dharma