There is a TIM store and Vodafone store both airside in Terminal 1 at FCO. Coming from T3, the TIM store has a large footprint in the connection hall outside of the stairs to exit and Vodfone is around the corner to the left, maybe 20m away from TIM.

There was no TIM or Vodafone elsewhere and all airport employees I asked all directed me to T1.

I chose TIM and started working from my laptop via my phone's hotspot on the train. TIM's reception was less than stellar throughout Roma, Napoli, Sorrento and the Amalfi coast.
I will be choosing Vodafone on my next opportunity.

Hope this provides any help to someone. Safe travels.