Can b12 deficiency cause shortness of breath even at rest and at time of bathing?

Possibly if anemic: B12 deficiency may be associated with macrocytic anemia. A condition where one has low red blood cell count. Blood cells carry oxygen from the lungs to the tissues of the body and carry carbon dioxide back from tissues to the lungs.

Not usually.: You could have shortness of breath with vitamin B12 deficiency but only if you are severely anemic as a consequence of the deficiency. The term pernicious refers to the fact that symptoms are usually minimal until very advanced deficiency is present. B12 deficiency is not specifically linked to breathing issues.

Depends: Long standing B12 deficiency may cause severe enough anemia to result in the symptoms you listed. If you are a vegan, you should take B12 and vitamin D supplements.