Hi, it could be due to nerve impingement or   some type of infection under the skin. Sometimes a very mild infection could trigger this type of sensation. If there is any inflammation or swelling in lymph nodes in that area of the body. It could create a sensation of pinching in that area. If a lymph node is swollen you could certainly feel it by just rubbing your fingers along the inside of your armpits. Consult Physician for further evaluation...