I'd suggest that the War Doctor would be more willing to carry a gun than other incarnations, but might not routinely carry one.
Evidence: as late as the events of "Dalek", the EccleDoc is more than willing to grab a gun and go to shoot a Dalek (it isn't until "The Parting of the Ways" that he fully repents his actions in the Time War ["Coward, every time"], thus allowing him to become new man, rather literally). On the other hand, Doctor Hurt (wasn't he a Batman villain?) has to borrow a gun from a soldier in order to blast a message into a wall in "The Day of the Doctor" - he doesn't have one himself.
Also in "The Day of the Doctor", the assembled Doctors sonic a Dalek to explosive death, so he might not need a gun. Or maybe you need three sonic screwdrivers at once to pull that off!
There is a licensed Doctor Who book, "Engines of War", featuring the War Doctor. Does anyone know if or how he is armed in that?