do waiters say "coming right up"?

Yes, but typically only in a more casual restaurant, such as a diner. It is not formal enough language for nicer establishment.

Worked 10 years in restaurants and I wouldn't say you're wrong but also I've never used the phrase myself. I think what you wrote certainly could be found in a dialogue, it just sounds a little Looney Tunes (cartoony) to me

No, but they do in movies.

It's a common phrase. Perfectly fine.

Only in an American diner maybe.

Yes, it’s a normal phrase that you might hear in a friendly, informal setting. I’ve certainly said it to customers (I work in a shop).

Coming right up as a response to an order is something a line cook might say in a diner. It would not be something a server would say; they'd repeat the order to make sure they got it right.

I've never heard it in my life in the context of a food order. I've heard it on TV or radio when they are talking about a programme that's going to be on shortly.

I feel like this would be a very normal response from someone actually cooking the food. Like a food cart seller, or just a small place where the person taking your order is also the one cooking it. Maybe a hibachi restaurant.

Yes, waiters say that. Is it proper academic English? Probably not.

I don't think that's a very good thing to say in that context. It will probably be at least several minutes before the salad is served. "Coming right up" implies greater immediacy than is warranted.

Your question has been answered, but I would also like to mention that most people in the profession prefer to be called servers. Waiter/waitress is an outdated term and can be considered demeaning.

More of a tv trope. I usually hear something like “I’ll get that right out for you!” (Southern US expression)

Unlike others here, I have heard it in work, when I was working as a waiter -- and not just from me. And yes... it was a diner. Well, a Jewish delicatessen/restaurant actually. I was the token goyim. :) Great job! They made the best sandwiches ever.

No. They would say “Ok” or “Sure!” and move onto the next person, or ask for specifications on your order. When they are finished with placing the order, a waiter might say “It’ll be out shortly.” or “Sorry, our kitchen is a little backed up right now so it’ll be a wait.” But “coming right up” sounds cartoonish.

I’m American and have only heard this on TV shows and in movies. I feel like it’s more common in media aimed at children than in stuff aimed at adults. It’s definitely a phrase that native speakers understand, so I don’t think it should be considered wrong. But it’s kind of old-fashioned and corny, so I don’t think you’ll hear it used much in real life.

That is a perfectly valid thing to say in that context, but if the dialogue was literally W: What can I get for you today? C: I will have a salad. W: Coming right up! That would be extremely strange, because the waiter would not know A) what kind of salad (most restaurants have several), B) what dressing (this may have been a vocabulary/content point the teacher would expect to be included if a salad was ordered) or C) if the customer wanted a drink (likely another vocabulary point) or any other items. The phrase "Coming right up!" would only ever occur at the end of an order, and ordering a salad does not sound like the end of an order to me.

Not in a UK restaurant they don’t.

In nearly every TV show where they are in a diner. I don't believe I have actually heard someone say it in real life, ever. That said, it isn't an unlikely thing for a waiter to say. I just don't spend as much time at diners as I should.

It's usually the kitchen staff that say that phrase, to alert the servers that the order is ready to be brought to the customer, or to alert the customer that their food is ready. It's not a phrase that the waiters would say to the customer. Maybe in some other places, that phrase is just copied to add "diner ambiance", but it's not for all used like that.