Yes, stress can cause itching. Stress-induced itching is called adrenergic pruritis, a minor variant of adrenergic urticaria, which is stress-induced hives/rash. Stress-induced itching can be reproduced by intradermal injection of adrenaline (epinephrine) or noradrenaline (norepinephrine).
Stress-induced itching can be treated with a beta blocker such as propranolol, as it blocks beta-adrenergic receptors, and prevents the binding of adrenaline to those receptors, thereby reducing the itching (Haustein, 1990).
Another way to reduce stress-induced itching is to reduce your stress and treat your anxiety.
Adrenergic urticaria and adrenergic pruritus. Haustein UF. Acta Derm Venereol. 1990;70(1):82-4.