My son used to stuff food in his face like a squirrel at home and school and then throw up when he was a toddler. He would always get sent home from school. I told them not to send him home because he was fine , he would just eat that way. They said it was policy to send a child home whenever he threw up. He would just eat too fast and stuff too much in his mouth. We had to teach him to slow down and not eat so fast. He is 11 now and 100% better.
When he was little he would also throw up when he got too emotional or upset. All his doctors told me it was anxiety. It took a while but now that he is older and can communicate better he has calmed down.

The gut is directly connected to your emotions so any fretting of any sort will affect it, whether you have autism or not. Imagine how overwhelming it is for them when you add in the inability to process the thousands of sensory input signals they have no ability to control. They need helps to compensate for the deficits.
My son would constantly tell me that his computer made his stomach feel better or "help him breathe". Now that may sound strange, but it is simply a tool to help transfer their focus so they can tolerate the outside stimulation. I assume it stabilizes the brain from overload. He was telling me what worked for him. Also feeding clinics will teach the method of watching TV to slow down the eating process. I have seen with that tool it slowed down his need to rush because he got more interested in what was going on on that screen. If you withhold the TV and use it as a reinforcer when he "does a good job", you may be sabotaging your effort because they will rush to get to the TV and thus not be able to process slowing down.Then they will throw it up due the anxiety of getting to the TV. So I used the TV to slow him down instead of a reinforcer. That worked.
With an inability to regulate themselves, they also often do not chew their food, because they overstuff their mouth. If you try it, you will see that to make room to chew, you must first swallow some unchewed food to make room just to chew what is left. It is very hard to digest what it not chewed. My son would throw up entirely unchewed food like he had no teeth. And he wasn't always stressed when he ate, he simply ate too fast and put too much in his mouth at one time. The pattern was 2-3 spoonfuls in a row without missing a beat before he even thought to start chewing.
Work on getting your child to take 1 spoonful, put the spoon down (hands off), chew long(maybe count, unless that bothers them), then swallow. Do not pick up the spoon for the next bite until finished swallowing. Yes, this is a tedious process, but if you don't turn it into a habit early on, it will be a battle later on. Research shows it takes 21 days to break a habit and 21 days to set a replacement habit. I wish I had known this when my son was growing up. You could also write the steps and let him see it as he eats.

My son does fine at all restaurants UNLESS it's over crowded and loud. The over stimulation makes him start to feel ill. He will immediately become red in the face and before we even get food he'll ask over and over can we go home now? Can we go home? I'm gonna puke, lets go to the bathroom NOW.
So if we go out and I see it's crowded I'll ask for a table away from everyone, no music speakers above us, also I'll take his iPad to watch a movie or let him watch YouTube on my phone so he focuses his attention on something else.
Also ice water not milk/pop/juice to drink.

Anxiety at mealtime has always been a problem for our 19 year-old if there is any emotional overlay. Could be related to who will be coming to dinner, where it will be, will there be anything she can eat (has to be gluten-free), is it different from where she thought we were going. Our best practice is to provide her as much detail as possible beforehand, such as: where and when, showing her a menu, calling first to confirm that certain foods are available, trying to help her see how her behavior can negatively affect others. Sometimes it can be triggered by smells of certain things cooking, or just a vibe she gets, thinking people are staring at her. Places that are used to kids and dealing with messes are best, even though the food might not be.

Thank you