Thing is, almost all of our sources have to do with people who didn’t disappear, so it's tough to say what would have happened. But at any rate, it would have been very difficult to vanish out of thin air. It is difficult to describe how much reliance a pre-modern city-dweller would have on their neighbors: You just would not be able to survive without regular human interactions. Especially for a person who, as you mention, is a merchant: in the 13th century, Mantova (or Mantua, if we want to adopt the english exonym) was an established stopping point in the florentine trade networks: Mantuan moneychangers were well-connected agents changing northern Italian coins for central Italian coins, and also managed letters of credit across the Apennine Mountains. Mantua also possessed a florid wool spinning industry, which required trade networks to purchase of wool from Tuscany and other parts of the mediterranean, and also networks to then resell them to consumers in places like Venice, Bologna, and the cities on the south bank of the River Po. And while Italian trade would reach its true apex a few centuries later, there is no doubt that even in the 13th century merchants in Mantua would conduct business reliant on a thick and fast succession of daily human interactions: while nowadays we might be used to being able to conduct business behind a computer screen and have many secure methods with which to identify our collaborators, no such convenience existed to help medieval merchants. Their activity was conducted exclusively in-person, and relied on personal networks built and maintained over a lifetime (if not over generations). An absence of even a day could mean missing a receipt of goods, or being unable to deliver goods to a client, and would be immediately noticed by collaborators and raise suspicion, be they business partners, agents, suppliers, or clients.

More banally, this is not only a time before the internet or even the mail-order catalogue, but this also is a time before refrigeration. Our hypothetical merchant would have visited three or four different shops every day in order to purchase food needed to survive. Their fishmonger would have expected them once a week, their baker would expect them to come in every other day with a bag of flour, their butcher likewise (depending on how wealthy they were), as would their grocer. Even more so than today, cities were dense networks of people constantly interacting with each other.

Additionally, they would have partaken in a number of social interactions: meeting peers at their guild, attending innumerable functions at the local parish church, or simply visiting friends and family. These informal social interactions are the lifeblood of urban life and a missing person would have been noticed very quickly.

We can add a few more notes from a criminal law perspective.

The increasing autonomy which the Italian cities developed over the course of the 12th century saw legal scholars dust off old books of Roman Law, such that the legal system came to increasingly resemble that which had existed a millennia prior. Some authors talk of a “Rediscovery” of Roman Law, but there’s no real indication that it went anywhere: medieval law had been an amalgam of the various varieties of germanic law, in which justice and lawgiving was administered by officials appointed by the monarchial authority (or rather, alternatively appointed or inherited) but there always remained a bedrock of old Roman Law to fill in the gaps. Prior to the twelfth century, the Italian communities didn’t always have the need or resources to apply Roman Law, as it required a minimum of urban institutional development (except in those places where it found ways to continue existing regardless, notably in Venice and Rome) but things changed over the course of the 1100s: As cities grew and professions specialized, jurists increasingly turned to roman-era laws (notably later roman - Justinian’s were popular) in the matter of penal law. This wasn’t a total shift, less than two centuries had passed since Otto’s conquest of Italy and German Kaisers still traveled to Rome to have themselves crowned Emperor, with not-infrequent conflicts emerging between Empire and Italian Cities, so when they could Emperors still attempted to redirect, dictate, and administer law in Italy, ultimately causing a “Mosaic” approach to legal scholarship (and the emperors themselves could lean on interpretations of Roman Law if they felt they needed to, just to keep things interesting for us).

So what does this mean in practice? This means that the gathering of evidence would be on the onus of the affected party, as is customary in the Roman Law system. In our case, if foul play was thought to be afoot, the “disappeared” person’s friends and family members would be tasked with gathering evidence and bringing them before the local “Podestà” (who could be a professional appointed by the local council, or a local notable who finagled themselves to authority) who would then appoint a judge to hear the case if they thought there was a case to be heard.

Who would our “disappeared” merchant have to rely on? Probably their spouse, but also siblings - oftentimes, shops or mercantile ventures were joint ventures between siblings, brothers-in-law, or even cousins; although cousins, more commonly, might branch out into associated businesses (e.g. one branch of a given family living out in the countryside, shearing wool, while the other branch in the city would take the wool to market. This is fundamentally how the famous Medici of florence built their early fortune, before branching into banking). Children were also close with their parents, with multiple generations often living under the same roof. So we can be certain that a merchant's family would have quickly noticed their disappearance and would dig into the matter. In addition, a disaffected party’s family could also rely on their guild. Even for individuals disinterested in politics (and few were - for the wrath of the mob was a near-constant in Italian communal politics) in addition to being the primary mechanism of political representation, the guild would offer a support network that didn’t stop in pure economic terms, but very much “Looked after their own.”

So there you have it. The length and breath of social interactions necessary for an individual, and especially a “merchant-type,” to exist in communal Italy would have meant that a missing person would have been noticed right away. But any investigation into their disappearance would have relied on the resources of their family members.