yes, there is some stuff to be had.
no, i doubt your naive friend would be able to find it.

first, understand that track cycling has been on the steady decline in italy. "fixed" culture is only a very small group of people, specifically the rotafixa chaingang. (you know, the no-lockring guys) there were lots of velodromes back in the day, but few are left. id start looking in those towns. (but first check when the velodromi closed)

second, in most urban centers you will not find bike shops specializing in road/track stuff. the city bike is the mainstay, if there is cycling at all. talking about bianchi, bottechia, crusers with full fenders and nice string spoke guards for womens dresses, with racks and pegs for holding packages and kids. you will only begin to see lots of old campy stuff slightly outside the cities, where road racers used to ride. riding thru tuscany, i saw more old guys on old bikes early in the morning than cars all day.

i cant recommend any track/road specific shops in rome but i would send you to the ciclofficina centrale, a squatted bike coop where members of rotafixa hang out, work on bikes, and promote cycling. when i was there as a broke student they had found a bunch of track frames for cheap and low flange campy track hubs to go with them - $25 euro a pair! maybe they have found alot more since, and if you ask nicely, they might point you in the right direction. more info at http://www.ciclofficina.org/