Ok, I know this is not for everyone, but at least you should give it a try!
The most famous street food in Florence is for sure the lampredotto, which is nothing but the forth and fifth layer of the stomach of the cow. If you read it like this it could sound not so good, but once you have tasted it onto a panino with the delicious salsa verde and some salt and pepper you could change your mind! The meat is boiled in a broth for a long time as to give full flavour to your panino.
My favourite spot where to taste the panino al lampredotto is right around the corner from my apartment, next to the Sant’Ambrogio church. Be prepared to stay in line during lunch time as that spot is really popular among local people!