In our campaign, we're using Vinto and Gideon. Vinto had his ability Shot on the Run not exhausted. Shot on the run allows you to exhaust it to deal a damage to something within three spaces and line of sight "during this activation" if Vinto has taken a move action. The question is, if Gideon commands Vinto to move, can Shot on the Run be activated?

I believe so, given that Vinto then has performed a move during "this activation," which just happens to be Gideon's. Most opinions I've found on the internet seem to agree with me. However, the rest of the players think otherwise, saying something about how actions taken during a command are not part of an activation according to some rule. Is there such a rule? And if there is not, does anyone have concrete evidence I can use to convince them? I've gone through the entire rulebook, and have not been able to find anything to prove myself wrong or right.

According to the RRG under Interruptions, interruptions pause actions, not the entire activation. So Vinto's move still happens during "this" (Gideon's) activation, so I'd say you'd be able to use Shot on the run until the end of Gideon's activation.

Note, Shot on the Run says "perform a move". It does not require the move action. Command does not give a move action, it allows to perform a move or perform an attack.

If Gideon Commands Vinto to perform a move, then Vinto has performed a move during this (Gideon's) activation.

So, because Shot on the Run does not say "during your activation", Vinto is able to exhaust it to activate the ability during Gideon's activation.

Note that due to the exhaust cost, Shot on the Run can be used only once per round (per activation token), but upto 2 times during the first round if Gideon activates before Vinto.

(When Order, Executive Order, or Command is performed, it is still the activation of the figure giving them. Activations are not split and only one figure can be activating at the same time)