There are some interesting suggestions here. We all seem to agree that it's not normal for a child to vomit like this and to be sick so often. She clearly has an immune system that is not where it should be.
I have a friend whose daughter had cyclic vomiting - but it was more frequent than your daughter's and more violent when it occurred. So they were pretty miserable for a long time, and there was little diagnosis or treatment available. Even so, the treatment is just putting a band-aid on the condition, not preventing it.
I know once a month is a lot for a kid to be sick, so even if it's not CVS, you need to do something. I think the "weird quirk" isn't - I think it's a key. Now, it's hard to know whether it's part of CVS or whether it's connected to her asthma or her tendency to get a lot of ear infections (which tells me there's a lot of mucus production and inflammation. She's not testing positive to food sensitivities but that doesn't mean she's not having some trouble with certain things. Still, that's not the entire cause because of the situations that seem to precipitate these episodes of vomiting.
I work in nutritional epigenetics - epigenetics is the field that deals with damage in the cells that effects the coating around our DNA (our genome). The DNA controls what our cells' function is, but the epigenome above/around it controls how well that function is carried out. So the genetic material is like your computer's hardware, but the epigenetic action is like the software - if there's a glitch in the software, the whole thing goes haywire and doesn't work properly. It can be a small switching problem that causes big problems.
What we're learning about so many diseases is that they relate to a small problem, which can be caused by a variety of things. But the discovery of, and extensive research into, a simple food peptide that corrects this damage, is bringing a lot of hope and documented clinical results to thousands of people and hundreds of problems. The interesting thing is that, because all cells have the same genetic material, they need the same nutrition. The genes in each cell that switch on and off are what determine the cell's function and appearance - which is why a nerve cell and a skin cell and cells in the stomach and lungs all look and do different things. A defect in the switching can occur due to disease, age, environmental triggers, and a whole host of other things.
In my experience, you can go through a long series of difficult and time-consuming activities like limiting intake of certain foods (then waiting 3 months to see if there's a difference), and then spend the rest of her life reading labels and restricting her in restaurants, friends' houses, and so on. It's not necessary. You can also medicate the symptoms (as you have with the asthma). You can go through something like the suggested detox of your home, but sometimes that stirs up as many problems as it eliminates.
But you can actually boost her immune system (and your own) very simply and repair the cells all through her body, which I would predict would solve the vomiting, the ear infections, the asthma, and all the colds.
This has been in well over 70 scientific papers and clinical studies, has been featured in major news magazines, on news programs and Dr. Oz, and everything in between. I've been to quite a few seminars on this subject, and the research is impeccable. Moreover, it's a plant-based solution which is totally safe. It's consumed by kids, pregnant women, people with cancer (it interferes with the spread of cancer cells), and everyone else. It's used in feeding stations and clinics serving seriously malnourished kids in a disease-ridden regions, and they are flourishing.
I have a lot of colleagues who are off their asthma meds, no longer have food issues, and have solved digestive problems like reflux and IBS. I myself am off all allergy and other meds, am no longer sick 4 times a year with colds and bronchitis, and am no longer dealing with clinical depression. If you want to try something else before taking this little girl through a whole bunch of diagnostic tests, I think it would be worth your time.