Throat lozenges and cold beverages can help to soothe the throat after vomiting. Irritation from the gastric juices are likely what is causing the sore throat.
Many throat lozenges contain medications that numb the throat. These lozenges also help to produce saliva that soothes the soreness.
Saliva and cold beverages help to soothe a sore throat because dry spots in the throat lead to more pain. Fluids also help to lubricate the throat, which can help to alleviate pain. If the pain is severe, throat sprays or over-the-counter medications might be necessary. A doctor should be consulted prior to taking these medications for a sore throat.