Well, the typical Chinese take-out container (also called oyster pail) that is used so often today wasn't patented until 1890.

This quote by George Robert Sims, a journalist who was born in 1847 and reported from London indicates that eating from reusable bowls at the stall and then giving the bowl back to be washed was the norm even after the industrial revolution.

Here in all its glory the ell-jelly trade is carried on. In great white basins you see a savoury mess. Behind the stall mother and father, sometimes assisted by son and daughter, wash up cups and spoons, and ladle out the local luxury to a continuous stream of customers. Many a time on a terribly cold night have I watched a shivering, emaciated-looking man eagerly consuming his cup of eel-jelly, and only parting with the spoon and crockery when even the tongue of a dog could not have extracted another drop from either.

There is also the painting Pepper Pot: A Scene in the Philadelphia Market painted in 1811 by John Lewis Krimmel depicting people eating Pepper Pot Stew at a street vendor's stall from reusable bowls. http://www.philamuseum.org/collections/permanent/134441.html

However, if mobility is important to your story there is a long tradition of naturally portable street food like meat pies, pasties, sausages, or meat, fruit or similar on sticks. Stews and the like could be served in bread bowls like it was common in medieval times.

It should be noted however that for a long time street food was the food of the poor since the quality of the used ingredients was often questionable and the poor were the ones who most often lacked access to cooking facilities. So depending on your setting and your characters they might not have eaten street food at all.