The basic requirement for a viable vegan diet is that it provides all of the Indispensable Amino Acids at the needed levels. When you read or hear a claim that "Rice and Beans is a complete protein", they are claiming it provides all these Amino Acids above a minimum level. There are many, many pairings that are sufficient, and countless more triplets, etc.

The first identifiable requirement is the development of agriculture. While it would be theoretically possible to survive purely off foraging if those wild plants were complete proteins, in practice, hunter-gatherers got much (if not the majority) of their caloric intake from animal meat. Furthermore, better preservation methods for meat (salting, smoking, drying) through the winter would possibly make it only seasonally possible. Thus, the next requirement is the ability to store plant based foods through the winter, necessitating a dry climate and an granary that protects the food from insects and rodents. This requires somewhat permanent settlement, with food spread throughout several granaries as an insurance policy. So the earliest a vegan diet was hypothetically possible was in the Fertile Crescent, Indus Valley, or Yellow River Valley, possibly as early as 10,000 BCE.

As for veganism existing within a larger movement, it is really hard to say. There are no written records prior to Cuneiform, and a ton of people other than myself would be able to speak much more about whether these records could even capture extent veganism and we could even recognize it if we found it. That being said, most discovered tablets remain untranslated, and most of those that have been translated probably could use multiple more interpretations, so we may discover more yet.

The earliest (7th Century BCE) hard evidence of at least vegetarianism, somewhat unsurprisingly, comes from the Indus Valley civilization. It was closely connected with non-violence towards animals, and continues unabated till this day. Its harder to say whether this involved veganism.

The modern rise of veganism, and veganism in other cultures is a much more daunting question, involving a variety of nuanced religious, political, and cultural factors. However, I will say that the truck and the tractor have played the largest role in increasing popularity, as people can now eat a variety of foods grown in different areas cheaply, it is a much more moderate hit to the amount of tastes in a diet than it once was.

Woolf, Peter J., Leeann L. Fu, and Avik Basu. “vProtein: Identifying Optimal Amino Acid Complements from Plant-Based Foods.” Ed. Niall James Haslam. PLoS ONE 6.4 (2011): e18836. PMC. Web. 2 May 2017.

Olivelle, transl. from the original Sanskrit by Patrick (1998). Upaniṣads (Reissued ed.). Oxford [u.a.]: Oxford Univ. Press.

Spencer, Colin. The Heretic's Feast: A History of Vegetarianism. 1993. Fourth Estate Classic House.

Pope, GU (1886). Thirukkural English Translation and Commentary. W.H. Allen, & Co. p. 160.

I can give a partial answer. Vitamin B12 is not available in a vegan diet and must be supplemented. It was first synthesized in 1972. It was not possible to be fully vegan until this time.

Edit: this may be incorrect. While in a wild diet, animal products are required to acquire B12, there may have been bacteria- or archaea-derived B12 made in a lab somewhere before the full synthesis was done in 1972, but this was still a 20th century activity. Edit again: further information: B12 was discovered in 1948. It was "produced industrially" in the 1950s but the source isn't clear on how. My guess would be bacteria-derived B12, but it may have been extracted from animal products? Not sure. Nowadays it's synthetic using the 1972 process.