OP - I went through this in the spring with my kid where every other week one of my kids contracted something.
It's awful for so many reasons: seeing your child suffer, the lack of sleep, the pressure it puts on your work etc. It's hard enough being a working parent with kids, and illness on top of it just throws everything out of whack. It can feel so overwhelming.
I also have emetophobia so that's a challenge too. But, like the others have said, there really isn't an alternative.
One way I try to refame this is by reminding myself that their bodies are strong and can handle it. I remind myself that this too will pass.
If your anxiety about this doesn't calm down, consider seeing a CBT therapist. They are good with stuff like this.