My dog is losing some of her fur?

There can be a lot of reasons for this. It could be a problem or nothing at all.

It depends on your type of dog and the season, but some dogs get extreme shedding once a year. Also if your dog is a golden retriever or has very thick hair like a golden retriever then it could be getting hotspots. Hotspots are just cuts that when moisture gets trapped in they could get inflamed. If this is the problem it's not serious, but there is a medication you can get prescribed from the vet if you want to to go away faster.

As for the throwing up, it could be just a coincidence. My dogs throw up a couple times a month, because they'll eat grass or something.

My friend's dog had cancer a couple years ago and the dog eventually lost all it's hair. If it really is a lot of hair falling out and the throwing up continues see a vet?

I've said all I know. Hope I helped and that your dog feels better:)