Emetophobia is unfortunately very common. (The reference linked starts with the DSM-IV definition of emetophobia. For those who cannot click through, it's a phobia of vomiting.) Before we get to the plethora of links, I want to strongly recommend a therapist, one who is familiar with the treatment of phobias and panic disorders. Any therapist who downplays the severity of emetophobia, or who may suffer from it themselves, will probably not be a helper. Most emetophobes exhibit panic episodes, like what you describe above for your daughter.
Like other phobias, however, it can be overcome. There are several different methods of treatment: I personally (and my daughter, 6) used hypnosis to overcome emetophobia. Others have reported success with EMDR, CBT (cognitive behavior therapy) and other types of therapy.
Please give your daughter internet hugs from me and my daughter. She's strong and can make it through this.