I cannot bear an unanswered question, so I'll kick this off and gladly hand over to one of the flaired master race.

There’s plenty of descriptions of foods as diverse as dried figs, sausages, fish and apples being sold in the street by a honking and blaring cacophony of street vendors, but not necessarily to be consumed in the street (although there's some evidence at Pompeii of sausage, spiced with pepper and pine nuts, being eaten on the move.)

According to City: A Guidebook for the Urban Age by PD Smith:

In Ancient Rome, there were vendors selling everything from wine by the glass to hot chickpeas by the plateful for as little as an ass (the smallest Roman denomination). There was such fierce competition between venders that food in Rome was cheap. In Pompeii, archaeologists have discovered some 200 bars, taverns and shops where you could be fast food, quite a number for a town of 12,000 people.

Roasted or boiled chickpeas (seasoned with cumin and salt) are a great example, as beans and pulses were pretty standard fare for lower income Romans to the point where the phrase friction ciceris emptor (“buyer of roasted chickpeas”) is effectively disparaging shorthand for the poor. As an aside, the poet Horace was almost defensive about the consumption of chickpeas, perhaps out of some early tinge of working class pride.

According to The Oxford Encyclopaedia of Ancient Greece and Rome, Volume 7, chickpeas were thrown into the crowd at major events like the Ludi Florales, a fertility festival that combined sporting games with “overtly sexual” theatrical performances.

Sources: Shopping in Ancient Rome: The Retail Trade in the Late Republic by Claire Holleran, The Oxford Encyclopedia of Ancient Greece and Rome, Volume 7 and City: A Guidebook for the Urban Age by PD Smith.