I'm a hairdresser. can the odor from hair dye cause an allergic reaction or the throat to swell/close?

Yes, tho...: ..it techically isn't an allergy. Could be hypersensitivity to the odor of the aerosol which can actually trigger tightness in upper airways such as laryngospasm or paradoxical vocal cord movement. Sensation usually triggers coughs but highly distressing airway spasm may develop. Sipping water and exiting area help. Can't rule out asthma so I'd see an airway specialist: allergist/ent/or pulmonary.