Vegan="素食主义者"， I suggest you say"我不吃肉"， it's simple and clear. Also, make sure if they use pig fat to cook. “请问你们用的是猪油吗？”

You can say "我吃素”

As far as I know, vegans are different from vegetarians or veggies. Thus, you'd better say, "我吃"全"素。" and then they wouldn't serve you meals with eggs, dairy food, and meat, etc.