What are the causes for numbness or tingling right under the right arm?

Couple possibilities: Numbness under the arm could be the result of pressure over a nerve, pinched nerve in the neck or arm, or even muscle overuse/strain due to poor posture. If correcting posture/repositioning does not resolve the concern, that warrants an office visit, particularly if there is an associated lump or mass, or any arm weakness.

Possible pinch nerve: It is not clear whether you mean chest wall or some part of right arm. In either case it seems disease of nerves coming out of neck in case of arm tingling or thoracic part of spine in case of chest wall tingling.