For a rice meal, you can say: Tôi ăn thuần chay. Xin làm đậu hũ với rau, dùng dầu thực vật, không nước mắm, không trứng sữa. That asks for vegan eating, tofu and vegetables with plant oil and no fish sauce, eggs or dairy. Then name the dish you want, so the kitchen has a real order to respond to.
For noodles, add: Nước dùng nấu từ rau hay từ thịt, cá? For a curry or dressing: Sốt có sữa, bơ hoặc nước mắm không? These questions are more useful than repeating no meat after a bowl has already been assembled. Let the person taking the order relay the complete request to the kitchen, including the dipping sauce.
Useful ingredient pairs are trứng/không trứng for egg/no egg, sữa/không sữa for milk/no milk and dầu thực vật for plant oil. Bơ can mean butter in a recipe but avocado in a fruit order, so give the dish context rather than showing an isolated translated word. For a mixed table, identify the vegetarian plate before the shared order is sent through. It is easier for the kitchen to prepare one clearly specified dish than to work out which diner should avoid which part of a large shared plate.