According to the oxford dictionary definition, a religion needs to actually worship something (like Buddhists for example who worship many deities with one central deity):
re·li·gion noun the belief in and worship of a superhuman power or
powers, especially a God or gods. "ideas about the relationship
between science and religion" Similar: faith belief divinity worship
creed teaching doctrine theology sect cult religious group faith
community church denomination body following persuasion affiliation a
particular system of faith and worship. plural noun: religions "the
world's great religions"
Although according to their second definition, which is hard to find an original source for other than it itself, it clearly would:
a pursuit or interest to which someone ascribes supreme importance.
"consumerism is the new religion"
The idea of only eating/benefitting plant based products is at least an "interest" or "pursuit".
The Webster dictionary has a similar variety of definitions:
religion noun re·li·gion
1 : a
personal set or institutionalized system of religious attitudes,
beliefs, and practices
2 a (1) : the service and worship of God or the
supernatural (2) : commitment or devotion to religious faith or
observance b : the state of a religious a nun in her 20th year of
religion
3 : a cause, principle, or system of beliefs held to with
ardor and faith
4 archaic : scrupulous conformity : CONSCIENTIOUSNESS
First definition depends on the word religious, which has as it's 3rd definition:
scrupulously and conscientiously faithful: FERVENT, ZEALOUS
It would definitely seem like vegans are faithful to their practice, in terms of being committed to it.
According to the 3rd definition of religion above "a cause, principle, or system of beliefs held to with", in order for veganism to be classified as a religion, it would need core tenants of belief.
To my understanding veganism is simply not eating or benefitting from animal products, without a reason provided. This could be out of personal preference for not liking the style of leather belts, to not finding meat healthy, to believing that all life that smiles at you should be saved but not the life that can't smile at you etc.
I met someone who claimed to be a vegan not because he believed it was better for animals in general, but simply because he personally didn't find meat to be healthy.
So if a clear universally accepted set of beliefs would be provided to describe all of veganism without debate, then it would meet most of the definitions from above, even if it doesn't involve the worship of a superhuman entity.