[matese]

If the vomiting is in the mornings when she first wakes up it could be from an empty tummy. Try feeding her a little food before bedtime.
As far as the rain her not wanting to potty, buy her a rain coat, carry her out, set her down and wait for her to do her thing.
When I use to leash walk my pack of 3, regardless of the weather, they were walked with the proper outdoor clothes. Rain, hail, sleet or snow, they were walked.

Your pup is controlling you, buy the proper outdoor gear for the weather.

I don't think she threw up because she choose to not go out, then again it could have from stress for not going out, it's hard to say, it could have also been just a coincidence that she threw up.

My guy uses the yard to potty, if he has to pee and sees it's raining he will not go out, 5 minutes later he's whining to go out and out he goes, pees fast and dashes back in the house.