I'm as concerned about an ear infection for a couple of months as I am about diarrhea for 7 days. Something is not right about this child's immune system. I work in food science and immune system boosting, and it does seem she is unable to get into balance. I'm not sure why antibiotics don't work for her and why you can't get in to see an ENT yet, but it's also possible that a lot of antibiotics have upset her intestinal balance so much that she's got this new problem. The vomiting can be a reaction to medication, or to a lot of post nasal drip (perhaps related to the ear infection and whatever else is going on in that area, such as a sinus infection, which can also relate to diarrhea). If she is teething on top of this, it's going to be difficult to sort it all out.
Meantime, she's not absorbing nutrition. Her gut is out of balance and probably inflamed. If she's insufficiently nourished, there can be vomiting associated with it (a similar thing happens in pregnant women who have such morning sickness they can't eat at all, then they vomit all the time).
I'd put her on the BRAT diet right now, in very very small doses. I'd give popsicles with a little real sugar in them (such as those fruit juice bars) or make my own, and I'd have her slurp on them throughout the day. Take her off dairy, put her on bananas, rice, applesauce & toast, perhaps try a small amount of peanut butter in very small amounts but frequently. There's an excellent children's supplement with very high absorption and a natural anti-inflammatory that can boost immunity and settle down that inflamed digestive system. It may help with the persistent ear infection as well.
I wouldn't wait for all these appointments, frankly. I'd be taking action at home and not be going to the doctor or thinking about the ER. All you're going to get is long waits, some talk about antibiotics (which they say won't work or which may have contributed to the problem), or some talk that her problems is viral and they can't do anything about it.