What is your aim in writing?
If you want your book to be a bestselling success, then you better delete that scene. The majority of people (as far as I can tell) prefer not to be sickened by their entertainment. Writing content like that will forever damn your book to a minority readership.
If, on the other hand, you write to indulge in your vomit fetish (for apparently you enjoy both the description and the actual throwing up, or you would stop writing it), that is, if you write for yourself, then by all means go ahead. There certainly is some interest in such material, as can be seen, for example, from the prevalence of porn that features vomiting (don't do a Google image search on that after a meal).
On a more general level, I don't think that the question is about courage, but about story. As a writer, you are not working on overcoming your disgust, but telling a story. Everything in your writing must submit to that goal.
Does it forward the story? Then it needs to be told.
Does it not forward the story? Then it needs to be deleted.
Courage can be found when a writer has a chance to educate the public by telling a truth the telling or publication of which will be painful to him or her.
What I guess I am seeing here, is a writer who lets himself get carried away. That's okay. Some of us like to write pages of steamy sex into our young adult romances, others go crazy with worldbuilding. But in the end it must all go. The porn sex becomes a chaste kiss, and the elaborate world is condensed to some hints here and there.
What I recommend is that you write whatever you like at the moment. Then, when you are finished, put your story in a drawer and let it rest for a few weeks or months. And then read it. You will have a clear feeling of whether this scene blends into your novel harmoniously or whether it sticks our like acne and needs to be healed.