My armpits itch. Is this due to an allergic reaction

Possilbly. Itching of the skin is a reaction due to some type of irritation. This irritation could be due to allergies, infections, insect bites, etc. Best to see your family doctor or dermatologist for an evaluation.
Maybe maybe not. There are many possible explanations for itchy armpits ranging from eczema, contact dermatitis (e.g. Deoderant) or underlying systemic disease. Visit your doctor for evaluation, perhaps basic labs and possible patch testing.