I guess if something went wrong with the procedure, maybe it could. My understanding is that the probability is low. I would check with a urologist just to be on the safe side. I know know my pied started well before I had a vasectomy.
They are not related to each other. Erectile Dysfunction can be a result of a rectal or prostate surgery, but vasectomy. It is unlikely that a vasectomy will directly cause Erectile Dysfunction.