Sicily Hotels - Do hotels offer a cot if you have a 3rd person in a room

Of course. But you'll probably have an extra charge.
You certainly cannot plan on having an extra person in the room unless you book for that person.

Are you booking directly with the hotel (I'd recommend doing that...not using a third-party site.?

It depends on the size of the room as well as how old the third person is. If the third person is a child, they may offer a cot at an additional cost. But if the room is small, there may be fire code issues with adding a cot.

Curious if cots can be requested when arriving

If you haven't booked a room for 3 occupants, I wouldn't count on 'adding a cot' on arrival. Suspect there are regulations on how many can occupy a room ( e.g., fire safety) or simply not enough floor space . You need to make the reservation for the number of bodies you intend to have in the room.

I've never had more than 2 people in a European hotel room, but if you do some preliminary screening on booking.com, it will ask you for the ages of any children in your party and then indicate which hotels have rooms that will suit you. Most of the European rooms I've stayed in simply had no space to add a cot.

You can request a "family room" they usually have at least 3 beds or the room may have one queen/full bed and one twin bed. I don't think I've seen rooms with two queen sized beds in Europe.

If the Sicilians in question learned British English at school you'll need to translate "cot" too. In British English that's what a baby sleeps in.

OP in a post about rental cars for Sicily mentioned '5 adults'

You need to book a room that sleeps 3. It is very unlikely that you would be able to add an additional bed on arrival.

We’ve stayed in many rooms that have an extra single bed on the side. Have you already booked your rooms? If so, I’d contact the hotel to assk if there are beds for three and if not, do they have cots.