The correct, profoundly unsatisfying, answer is: we have no idea.

But the absence of evidence, as historians like to say, is license for rampant speculation. So let's indulge...

Since Vesuvius probably erupted in late October, near the end of the sailing season, it is fairly unlikely that any merchant would have been returning from a long-distance voyage at the time of the disaster. But let's say some bold soul decided to brave the uncertain seas of autumn for a quick run to Africa. With favorable winds, the voyage from Carthage to Puteoli (the great port on the Bay of Naples) took only a few days; and perhaps our fearless merchant's return trip was hastened by a roaring Auster (sirocco) gale.

Since the trip from Carthage to Puteoli was over open sea, there would have been no opportunity for news of the eruption to reach the ship. The crew might have noticed that the sunset was unusually spectacular. But since the plume of Vesuvius' debris paralleled the line of the Italian coast, they would likely have been unaware that anything was amiss until their ship rounded Capri and entered the Bay of Naples.

Then, suddenly and terribly, they would have confronted a vast stretch of coastline transformed into a smoldering wasteland. As the ship drew close to where Pompeii should have been, rafts of pumice rasping against the hull, the crew would have struggled to recognize even basic landmarks. The pyroclastic flows that buried Pompeii had driven hundreds of yards into the sea, burying the old harbor and its quays. The city itself was almost completely covered, with only the tops of a few battered buildings emerging from rubble.

Unable to land along the jagged cliff of the pyroclastic flow, the merchant would probably have continued to Naples, the closest intact port. The harbor would have been crowded with the vessels from the fleet at Misenum and boats that had fled from Pompeii. Once the merchant finally found a berth, the first person he encountered on shore would have told him about the disaster. It is unclear, however, whether passers-by or officials could have helped the merchant find his family.

We don't know whether there was anything like a refugee camp for the survivors. The emperor Titus mounted a swift response to the disaster, but Suetonius (Titus, 8.4) tells us only that the emperor chose commissioners to coordinate the relief effort and made funds available for the reconstruction of the lost cities. Nero had shelters set up for victims of the great fire of Rome, and allowed refugees to sleep in public buildings (Tacitus, Annals 15.39). Perhaps Titus provided similar relief to the victims of Vesuvius. If nothing else, it is likely that there were informal gatherings of Pompeiian survivors around Naples and the adjacent towns, but we have no idea whether the marines stationed at Misenum set up formal camps for them.

It is more likely than not that merchant's family survived the eruption, since most of the city's inhabitants managed to escape during the early stages of the disaster. As far as we can tell, about 2,000 people died at Pompeii - perhaps a tenth of the total population. But it probably would have taken the merchant some time to find his family in the refugee-crowded city, particularly if there were no organized camps.

All of this, as I said, is speculation. The sources are silent. But I hope this provides at least a general sense of the chaos that surrounded the disaster.

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There are plenty of books on Pompeii, but the most accessible one is undoubtedly Mary Beard's The Fires of Vesuvius. There are also plenty of novels (engaging, not necessarily edifying) centered on the eruption of Vesuvius, most recently Robert Harris' Pompeii.