Prior to 2023, I would have said "no" and that you can only buy tickets at edicole, tabacchi, and in the Metro stations.
However, in the past year, ATaC has put contactless card readers on each of their buses, trams, and Metros (at the turnstiles) so you can now pay by card on board.
While it's still helpful to have a physical ticket, just in case you don't trust the contactless card systems, you no longer need one!
For more on ATac's transportation systems, you can find my article here.