The time of year which blueberries are grown depends on the area in question. On the East Coast of the U.S, blueberries are grown mostly through the summer months. While it is true that blueberry bushes are grown year round, that is not true for most of the east coast.
Blueberries start to ripen as the weather changes and become warmer, starting in the month of May. While it is certainly possible to pick blueberries in other seasons or times of the year, they usually start to get into grocery stores in the winter; they are stored in temperature-controlled rooms from the summer and then shipped out to stores around December.