Okay, so you're definitely going to need:

Tôi ăn thuần chay (Toy uhn tw-one jai) tw-one as if you were halfway to saying two before changing to the word one very quickly, and jai rhyming with sly. This means "I am vegan".

Xin một [insert food/drink] thuần chay, e.g. "xin một bánh mì thuần chay" (Sin mop bahn mee tw-one jai) - "mop" is said with a stop on p sound. You know when we say words like hot or heat how we don't usually breathe into the T at the end but we just come to a sort of stop, where it's almost a T and a D mixed but we have no emphasis on the last bit; we don't breathe into the T and say "hea-tuhh" like people speaking with upperclass British accents do? That's how you should say "Mop". The P should be basically silent instead of "Mop-puh", if that makes sense. No air should escape your mouth when you say the P.

Anyway, that's how you say "Please, one [dish] (made) vegan". So if you go into a restaurant and say that you're vegan they will probably understand. Then say "xin một [food dish] thuần chay" and they'll understand because you're priming them with information. You can also point at the menu or a picture instead of saying the food name. I'd confirm with an "Okay?" to get them to confirm. If they say no then you can try again or negotiate with them from there.

But please make sure that you listen to them when they repeat the order back you because "chay" just means vegetarian. So if you hear "Bánh Xèo Chay", make sure to correct them with "Bánh Xèo Thuần Chay".