Are there any Muslims in Vatican City?

The population of Muslims in Vatican City in 2010 was around < 1,000 comprising approximately < 0.1% of The population of Muslims in the country.

The projected population for 2030 is around < 1,000 comprising approximately < 0.1% of the country.

Source: Pew Research Religion and Public Life Project

There is currently no mosque in Vatican City. There is no synagogue or pagoda either. There is not even an airport (there is a heliport), stadium (even though they have a national soccer team), kindergarten or school (even though there are kids), or McDonald's (people in Vatican City need to eat too). For the most part, they have access to those facilities outside the city walls, in the territory of the Italian Republic.

It should be noted that there is no ban on building mosque or pagoda or stadium. The real problem, however, is that you can't actually find a plot of land to build it. Vatican City is basically comprised of a tiny walled estate, and nothing else. The use of land has already been highly optimised, due to the limited building area. Moreover, there is no public land available; all lands are private property of the Holy See (which is a distinct entity from the Vatican City State), and none is for sale; with the extremely lack of land for their own use, I don't think the current owner is willing to transfer ownership unless there is a strongly compelling reason.
Last but not least, a mosque is obviously not a top priority in Vatican City, as there is practically no Muslim resident inside its city walls for it to serve. Muslims residing in Rome, on the other hand, are served by the Mosque of Rome, the largest mosque in Europe (which is about 6km from St Peter's Basilica).

So, it can be said that assumably, all the Muslims in the Vatican City are just tourists who have come to visit it and virtually, no Muslims reside in the Vatican City.