I'm aware that Theravada, Tibetan and Chinese Buddhism have a canon of texts. However I have never heard of a canon for Zen Buddhism. Does it have a canon or does it just not bother with such a thing? If it doesn't have a canon of texts then are there texts which can seen as foundational?
Zen Buddhism is part of East Asian Buddhism in general and accepts the entire Chinese Canon, although they don't find a lot of it to be relavent to their practice. Zen Buddhism has always been based primarily off of the meditation practice itself combined with oral teaching, but even if Zen isn't purely derived from the Sutras in the way other schools are, the Sutras are still a very important part of Zen learning.
Originally the main text for the Zen school was the Lankavatara sutra, but as time went on, this very difficult Sutra became less popular and the Diamond Sutra became a much more central text, along with the Heart Sutra. The Platform Sutra which actually tells the story of Huineng, the sixth patriarch of Zen, and how he rose to prominence also became a primary text.
A treatise called the Awakening of Faith in the Mahayana was very influential, and the Sutra it was based on, the Sutra of Perfect Enlightenment also had a good deal of influence, although it isn't quite as famous these days.
Next in importance to these canonical texts are the two famous Koan collections, the Gateless Gate and the Blue Cliff record, at least in the traditions where Koan is a major focus. For the Soto Zen School which began to de-emphasize Koan practice, the great classic is the treatise called the Shobogenzo, written by Dogen Zenji, the Soto Schools founder.
Those are all the major zen texts I could think of.
(Added in later):
There are a few others I forgot about. All of the Sutras that are about Buddha nature (The Tathagatagarbha Sutra, the Mahaparinirvana Sutra, the Srimaladevi Sutra, and especially the Lotus Sutra were very important in Zen. Also, the Uttaratantrashastra was important although it's a treatise, not a sutra. From very early on, the main slogan of Zen was "See mind, become Buddha" which meant that by directly meditating on the mind, one sees the Buddha nature of the mind and becomes enlightened.
The Gateless Gate. The Heart Sutra. The Diamond Sutra. The Shōbōgenzō. These are the first ones that come to my mind as a Zen Buddhist.
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