The Vandals and Goths ruled Rome for several years after the fall of the empire. Did they leave any relics, monuments or buildings that can still be seen?
As far as I know, the Vandals never established themselves in Rome. They settled in Iberia first, and were later driven away to modern Maghreb by the Visigoths.
They set up a fleet, which was instrumental to their attacks in the Western Mediterranean Sea, culminating with the sack of Rome of 455 CE. After pillaging the city for two weeks, Genseric and his men safely retreated to Africa.
Similarly, the Visigoths never really exercised any formal authority in Italy. Despite occupying the peninsula for several years, they only settled in modern days Southern France.
On the contrary, the Ostrogoths, which occupied Italy on request of the Eastern Roman Emperor Zeno, exercised a stable, if short, kingdom. Their kings, most notably Theoderic the Great, sought a pacific coexistence of the Roman and Goth ruling classes. The civilian administration was left to the Romans, while the Goths firmly controlled the military.
Theoderic is famous for the Basilica of Sant'Apollinare Nuovo, and most notably for his Mausoleum, both in Ravenna. This is the most distinctive Goth monument in Italy, built with a "barbarian" style which for instance did not employ bricks (unlike the Basilica, of Roman style). The Ostrogoths were mostly based in Northern Italy, as witnessed by the fact that Ravenna was the capital of the newborn kingdom.
Theoderic and his daughter Amalasunta funded several maintenance works in Rome. Although I cannot exclude that the Ostrogohts left some monuments in the older capital, it must be noticed that this city was no longer a centre of political nor economic power, and if any monument have been built there, they are unlikely to be as important as the Mausoleum.
Sources: memories from various works, most notably "The History of the Decline and Fall of the Roman Empire", E. Gibbon and "History of the Byzantine Empire", G. Ostrogorsky.