Can one embark upon a hair transplant with a permed hair or one must wear her natural hair?

You can perm your hair before or after surgery.  Hair transplant involves harvesting hair follicles under your skin.  

Yes, you can perm your hair before surgery. It will not have any affect on the results,
Dr. Robert Jones

I tend to tell my patients to avoid it.  The reason being most that the permed hair might be difficult to work with during the procedure and it's best to have a clear vision of the area we are transplanting into to assure the best result.