Purchasing, seizing, or being gifted one of the smaller caribbean or atlantic islands seems possible, even with very little divergence from OTL. Perhaps a very catholic and very wealthy ruler of Spain gifts a sugar island to the Holy Father on his birthday or the anniversary of the completion of the reconquista, and in gratitude for the Pope's affirmation of the Spanish-Portuguese personal union.
More substantial colonies would require a more substantial papal state. Have one of the Popes successfully reform the feudal mess into something governable, and then expand in Italy. In the 15th century the Pope granted Naples and Sicily to Aragon, perhaps Naples could have instead been annexed. Florence and some of the other small Italian states were occasionally de facto vassals of the Pope, these could presumably have been formalized, especially if the relationship between the Holy Roman Empire, Italy, and the Papacy changes.
With Italy under Papal control by 1600-1700 they're in as good a position to colonize as the Netherlands, and better than OTL Germany. They could establish forts on the African coast, as bases for military and commercial navies and for missionary work. They might get Patagonia or some islands, anything else in the Americas seems unlikely unless with a very early POD and largely unrecognizable world. They could eventually establish further bases in SE asia. I expect their colonies further from home wouldn't be very large, and they wouldn't try to compete with more powerful states for control of places like India or Australia.
With improved medicine and the colonizable parts of the world running out around 1900 we might see some larger Holy Protectorates. Rome is in as good or better position to take vast swaths of the Sahara as Paris is. The previously mentioned bases will be the start of larger colonies in Sub Saharan Africa, and the Papacy might get additional benefit from Catholic missionaries based in other state's colonies claiming land on behalf of the Pope rather than their monarch.
Finally, perhaps if a colonizing nation converts to a different religion than catholicism, and then loses a war, perhaps they might have to cede some of their colonies to the Pope. A protestant Portugal giving Brazil to the Pope because Spain, France, and England don't want each other to have it would be interesting.