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.
Here is more information:
I am from a Chinese Buddhist organization with Chan lineage but is non denominational. They use the entire Chinese Mahayana canon as their basis, but in practice only popular passages like the Heart Sutra, Great Compassion Mantra, Amitabha Sutra, Universal Gate Chapter of the Lotus Sutra are chanted on a regular basis by lay devotees like myself.
To understand the purpose and vision of the Chan/Zen school you must go back to it's Mahayana origins, which arose out of historical Buddhism due to it getting out of touch with the masses, Buddhism itself was on the verge of dying out. As a result immense amount of relatively new sutras and scriptures were created to explain the vision of Buddhism and application of the dharma to the world, using fantastical stories telling with hidden meanings. However the teachings themselves eventually become a baggage. Instead of practicing the Dharma, monks can end up spending time trying to absorb all the content. Many end up debating philosophy instead of practicing. Many temples become purely ceremonial instead of being a place of practice.
Further more some schools end up having very elaborate rituals requiring great expenses making it difficult for the poor to access Buddhism. Confused, undisciplined monks behaved badly, the public lost faith.
So Chan/Zen was a 'back to basics' poor man Buddhism, with focus on mindful concentration attaining the Jhana (aka Chan/Zen) and applying it to all situation. They farm, but farm mindfully. Cleaning, eating, walking and of course sitting meditation are expected to be done mindfully. Study of sutras were de-emphasized for practice instead.
I highly recommend watching the movie Zen (2009) about Zen Master Dogen and the vision behind his school. When he returned to Japan from China he found the practice have degenerated and monks behaving badly and set out to revitalize Buddhism. His rivals were outraged, did he bring back new sutras? What is this true Buddhism he spoke of? But the simple fact was that without practice you could never truly reach the truths of Buddhism. And even enlightenment was not the end of practice.
As the Buddha himself said, once you crossed the shore of suffering, you don't need to keep carrying the raft/teachings with you, because you would already know what to do.
Hence, all the knowledge of the sutras, while important in the cultivation of wisdom, are no substitute for practice. Buddhism is not truly Buddhism without the Noble Eightfold Path.
You can see similar historical trends in the forest tradition of Thailand relative to city temples.