I've had puffy nipples for 4 years. I'm 16. What are my options?

At your age, the most likely issue is that your breast development is not yet complete.  We stage breast development using Tanner staging.  You are born at stage 1 and fully mature is stage 5.  The only difference between stages 4 and 5 is that at stage 4 your areolae (the pigmented part around the nipple) is raised off the breast mound.  Give yourself more time, while some women stay at Tanner 4, most go on to Tanner 5 and have flat areolae.