So although I'm more well-versed in the trade of finished goods rather than the specifics of the grain trade, I can speak towards a number of trends in urbanization which might answer your question.

The short answer is that ancient and modern urbanization in Italy were two different things. In antiquity the general trend was for cities in Southern Italy to be larger. These cities, in the dead center of the Mediterranean, were entrepôts integral to the Mediterranean economy; so much so that they were majority greek-speaking! Although parts of Southern Italy were indeed very fertile, like Puglia, Sicily, and the Campania, Southern Italy's disproportionate urbanization was certainly dependent on trans-mediterranean grain shipments, and these cities declined as a result of that trade's disruption.

Italian urbanization in subsequent centuries was mostly focused on the northern half of the peninsula. Characterized by a complex hydrography consisting of a series of parallel waterways flowing down the Alps across a broad plain to Italy's longest river, the Po, Northern Italy could support dense urbanization without relying on grain shipped from afar.

However, it's important to understand another feature of Italian cities: although life in Italy was indeed characterized by urbanocentric government and society, most Italian cities were actually medium-sized. Venice, for example, would be completely unrecognizable to us as late as the 14th century; resembling a series of islets spread over hundreds of square miles of brackish water lagoons between the estuaries of the rivers Sile and Piave. Indeed, the Venetian census of 1338 lists only 40,100 residents. Of course, Venice is a notable outlier in all things having to do with Italy: in the Honorantie civitatis Papie three centuries prior, Emperor Otto's administrators marveled that the people of Venice "Do not plough, do not sow, and do not harvest" but never seemed to suffer from want of food. Plus, more than one economic historian has complained that Venetian sources themselves are surprisingly lacking on the topic altogether! It would seem, then, that Venetian commercial penetration in the Northern Italian interior was more than sufficient for the city's food security needs; and besides, records of Venetians trading salt and garlic for grain in the Friuli, not too far from Venice, indicates that the North Italian plains were fertile enough to supply the city without merchants having to travel too far inland. By the end of the 14th century, there are records indicating that grain merchants, like Pietro Regla and Lunardo dall'Agnella, had enough wealth to pay the fees associated with to joining the city's aristocracy, however Venice at the time had just come off a long and taxing war with Genoa; it's unclear if the wealth of these individuals is tied to a hungry population or mere war profiteering.

Venice's true demographic explosion occurs in the 15th century, when the city took advantage of a chaotic political situation on the mainland to construct a large state outside the lagoon. The Republic's expanded borders allowed for food security to sustain accelerated settlement in Venice, and the census of 1422 registered 190,00 inhabitants.

Most Italian cities of some relevance (Pisa and Siena, but also Verona and Bologna) are estimated to have had around 50,000 inhabitants, with the true protagonists (like Genoa, Florence, and Milan) being more or less twice as large. Of these only Milan, in the dead center of the North Italian plain, registered a truly massive population estimated to have reached 100,000 residents as early as the twelfth century. Florence grew more slowly, starting from an estimated 30,000 in the twelfth century to reach 90,000 in the 15th. Florence is south of the river Po's floodplain, so Tuscany's hummocky terrain, while not as rugged as much of Southern Italy, is not well-suited to intensive agriculture. Thus Florence's growth is predictably conditional on its growing area of territorial control and political relevance, eventually organizing a dole purchasing grain from across the Apennine Mountains in the Romagna to be resold in the city at low prices (this policy perhaps influenced Florence politicians' attempts to keep Papal control in Bologna weak). But in any case, the point is that agriculture on the Italian peninsula was the main contributor to food security.

I think Genoa, in the thirteenth century, was most reliant on far-flung grain-fields; pressed between steep hills and the sea, Genoa's region of Liguria is certainly not a candidate for intensive agriculture. Indeed the Republic of Genoa's government pushed its zone of control increasingly northwards, over the Ligurian Apennines, in order to secure areas suitable for agriculture (that's why there's a town called Novi Ligure in Southern Piedmont) until they came to rely on commercial links to Northern Europe for its own grain dole. However, by the fifteenth century the Republic became so commercially and politically linked to the Milanese state to the north that it acted as a de facto client, securing access to produce from the North Italian Plain.

So in short, although food security was certainly an issue, Northern Italy could count on its own production to meet its needs.