What was Tenpenny looking for there? Dukov mentioned it was some type of gun but I never found out what exactly.

You might get more of the background story by talking to a certain Ghoul in the Underworld.

http://fallout.wikia.com/wiki/You_Gotta_Shoot_%27Em_in_the_Head

The wikia says Tenpenny hired him and the others to get a rare weapon. I wanted to know what that weapon was. (Already got the armor).

No, not a gun. A suit of power armour.


Then you already have all there is. The Wiki is wrong, there's no special weapon.

Oh, and it's Fort Constantine, not Fort Bannister. Fort Bannister is where the Talons hang out.

Tenpenny hired them to get a weapon, not the armor. Jeff Strayer, Dukov, and Dave apparently got paid for the job when they returned, so I can only assume they had acquired it. What that weapon was, I have no idea.