There are so many options to eat. A few of my favorite street food items are Islak Hamburger (a saucy hamburger), Ciger (liver, actually), Midye Dolma (stuffed mussels), and Kokorec (like a spicy sloppy joe, with slightly grosser ingredients).
Check out these sites, and blogs for more on street food:
Also, check out these posts describing Turkish food in more detail:
If you want to have different street food experience, go to Kilisli Fiko. It is in Kadikoy but not in the center. You should walk towards Haydarpasa hospital.
All over town, after midnight you will see old-style Volkswagen buses selling meatballs. You can find them near the Bosphorus, ferry stops or stadiums (if there is a game, of course).
Definitely check out our blog post about it.
One of the most famous street food vendors on Anatolia side, is Asker who brings his food cart to Kadikoy's Rihtim Caddesi (Rihtim Street) at late hours of the night. He can be found near Murat Muhallebicisi (restaurant). His speciality is cold sandwiches.