Bible historicity is a complicated question. First, we don't know where exactly the cities are supposed to have been - the Bible itself is somewhat vague and there's never been a consensus as to their exact location since. We therefore are unable to point to any archaeological evidence as proof of their existence or as a way of working out what the story is based on. They may have existed, they may have not.
It's important to note that the historicity of anything at all in the Pentateuch (the first five books of the Bible) is heavily contested. Certainly these early stories about Abraham and Lot would have been written down hundreds of years after they happened, if they did. Israel Finkelstein is an example of a biblical minimalist who in his book Unearthing the Bible argues that the books were probably written down at the earliest in the era of King Josiah (640-609BCE) and contain many anachronisms (such as camels nearly a thousand years before they were brought to the region) which indicate heavy editing, even if they may have been based on folk tales or traditions. Indeed, some stories in Genesis barely make an appearence in books theoretically written later even when they seem relevant, raising the possibility of an even later date of writing for these. However, Sodom and Gomorrah does appear multiple times in other books - for example, Ezekiel 16:49-57, Jeremiah 23:14 and Amos 4:11 - indicating it was a well known and old story. Indeed, according to Claus Westermann in his 3 volume commentary on Genesis, "There is no event in the whole of Genesis that is mentioned so frequently in the Old Testament as the destruction of Sodom"
A common way of understanding stories set in particular locations is as etiological - they give an origin story for a particularly notable place or as a way of explaining their name. In the Bible this has the added benefit that each part of the landscape explained this way can be used as "proof" of God's actions. There are multiple stories where a name is given an etymology based on a narrated story - for example, in Genesis 32:31 Jacob names the place where he wrestles with the man/God Peniel "because I have seen God face to face and survived" (El is a Hebrew word for God). The names certainly proceeded the stories and the etymologies given are invented.
The particular story of the destruction of Sodom and Gomorrah has often been seen as etiological - either there were genuinely ruins of cities which the story was attached to to explain how they came to be or it describes a particularly distinctive landscape - probably somewhere around the Dead Sea - that was explained by the rain of fire and brimstone. The Jewish historian of the 1st century AD Josephus even claims to have seen the pillar of salt Lot's wife was transformed into!
In the other parts of the Bible in which the story is referred to, Sodom and Gomorrah act as the archetypal cities destroyed by God's judgement and the prophets often use it as an example of what would happen to the Israelites if they didn't follow God's laws. Each time it is brought up the crimes that it was destroyed for - and therefore the crimes the Israelites are being called to repent for - varies. In Ezekiel 16:49-50 (NIV): "She and her daughters were arrogant, overfed and unconcerned; they did not help the poor and needy. They were haughty and did detestable things before me." In Jeremiah 23:14, the prophets of Jerusalem "commit adultery and walk in lies; they strengthen the hands of evildoers, so that no one turns from wickedness; all of them have become like Sodom to me, and its inhabitants like Gomorrah." Claus Westermann suggests that the Genesis story was originally based on the tradition of these two cities destroyed - with Genesis 19:24-25 being the base story simply of God destroying the cities - with a whole new story based around Lot and Abraham being weaved around it, telling a particular theological point. The Genesis 19 story focuses on hospitality and makes a contrast with Genesis 18, another hospitality story (as pointed out by Thomas Brodie in Genesis as Dialogue) - in 18, Abraham hosts a divine trio as a perfect host, leading to divine reward (Sarah's child). In 19, the lack of hospitality of Sodom leads to the ultimate divine punishment (total destruction) and Lot, as the only person to offer hospitality to the visiting angels, is the only one spared.
To sum up, we can't know any historical facts about the cities as we only have the Bible stories to go on. There was likely an old tradition about Sodom and Gomorrah being evil cities destroyed by God that many biblical writers drew on. When writing Genesis, the author of the story likely used that old tradition and weaved it into the broader Abraham narrative, telling his own particular story about God, mercy and repentance.
If you're interested in Bible historicity, The Bible Unearthed by Israel Finkelstein is a good accessible book on the subject, but keep in mind that it's giving one particular interpretation of the evidence that's called the "minimalist" school and there's alternative viewpoints. If you're interested in an introduction to how the Bible is interpreted, both historically and using modern academic biblical criticism methods, James Kugel's How to Read the Bible is an entertaining and highly accessible book.