Could getting a tattoo or piercing while breast feeding affect the milk?

Yes. Make sure to go to a clean and sterile studio that follows universal precautions, including using an autoclave (that is spore tested at least once a week), single-use ink caps, gloves, and needles.
Nope. Tattoo artists will not do it. You have to have your "6 week clearance" after you stop breatfeeding from your doctor. If you get a tattoo while breast feeding, it CAN possibly enter the blood stream and then pass through the breat milk. It is just not worth endangering your child if something MIGHT happen.