Is there a way to make my nipples less puffy? (Photos)

Your areolas are puffy because you have mildly tuberous breasts where some of the breast tissue is being pushed out through them. Surgery is the only way to correct this with a scar around the areola to reduce the size. 

Dear nippgirl, yes there is!  Unfortunately it involves surgery.  Your nipples are actually normal, what is puffy is your areolas.  What is happening is that the areolar skin is much weaker than your breast skin so the glandular tissue of your breasts pops out into the areolar skin making them so prominent.  That is part of what is known as constricted breasts.  Your breasts are nicely shaped and pretty but the areolar feature is not of your liking.  The way I know to correct this is to reduce the diameter of your areolas via a donut reduction.  In doing that there will be less of an opening for the breast tissue to protrude into the areola.  The scar will be a ring at the very edge of the areola and therefore concealed by the color transition. Hope this helps.