Yes, stress can cause skin rash. Stress can induce the development of hives, or urticaria, a type of skin rash that looks like raised, pale red, itchy bumps. Stress-induced hives is also called adrenergic urticaria, as the rash can be reproduced by intradermal injection of adrenaline (epinephrine) or noradrenaline (norepinephrine) (Shelley and Shelley, 1985; Hogan et al., 2014).
Besides stress, adrenergic urticaria can be triggered by coffee and chocolate. The treatment for adrenergic urticaria (stress-induced hives) is to deal with your stress and avoid triggers. In addition, beta blockers such as propranolol can block the adrenaline from binding to beta-adrenergic receptors, thereby reducing the rash and itching caused by stress. Other treatments for adrenergic urticaria include benzodiazepines and antihistamines (Hogan et al., 2014; Kawakami et al., 2015).
Adrenergic urticaria: a new form of stress-induced hives. Shelley WB, Shelley ED. Lancet. 1985 Nov 9;2(8463):1031-3.
Adrenergic urticaria: review of the literature and proposed mechanism. Hogan SR, Mandrell J, Eilers D. J Am Acad Dermatol. 2014 Apr;70(4):763-6.
Refractory case of adrenergic urticaria successfully treated with clotiazepam. Kawakami Y, Gokita M, Fukunaga A, Nishigori C. J Dermatol. 2015 Jun;42(6):635-7.