Italian restaurants--literally. In italy.
I second Volpetti. I went there last summer a couple days before leaving Rome and the owner was there - when I said I was looking for gifts to take home, he started having me taste just about everything in the store... olive oils, balsamic vinegars, hot pepper jam, cheeses, meats - everything. (Of course, then I felt compelled to BUY it all...!) The owner is incredibly proud of having been featured in an american culinary magazine and handed out a photocopy of the article with my purchase. In addition to vacuum packing EVERYTHING, he also gave me Volpetti paper, ribbons and stickers so I could wrap all the gifts!
My favorite place for a huge, cheap, satisfying sandwich in Florence is Antico Noe, which is hard to find - in an arcade a couple blocks away from the Duomo - and has no seating. Stand in the alley with your sandwich and glass of wine with the other locals and students.
Recent posts on Rome seem to have that covered!
In the 30 seconds before this post gets knocked over to the Italian board, I will make the obvious suggestion that all your food-smuggling needs can be taken care of at the Volpetti store in the Testaccio nabe of Rome. There is a stunning selection of unimportable artisanal meats from all over Italy, especially from the pork Mecca of Norcia, amazing cheeses, and the store vacuum wraps well enough to fool dogs. Have 'em wrap it twice. You may be able to find marginally better examples of Tuscan pecorinos, real culatello and lardo di Colonnata closer to the sites where they are made, but I suspect you may not.