I personally made mine an Armour Storm - rolling him as a Vesk and carrying around (currently) a laser rifle....Loved the idea of comboing the power of Hammer Fist with the Vesk's eventual 1.5x str bonus to it, especially since I'm planning to pick up the Melee Striker gear boost at level 3 - so he'll have a total of 1d4+6 damage by that point.
I've also created an armor storm Vesk, and I'm excited about playing him, but sadly your beautiful dream of combining Hammer Fist and the 3rd level Vesk natural weapons specialization is not to be. The description of Hammer Fist makes clear that "these unarmed attacks don’t benefit from other abilities that apply specifically to unarmed attacks."
If that's not enough, the Vesk racial description states that the 3rd level specialization lets them "add 1–1/2 × their character level to their damage rolls for their natural weapons." The wording there strongly implies that this bonus is for attacks made with their bare fists/head spikes/tails/etc., not with armor-clad body parts.
I mean, hey, if your GM will go for it, rock out. But my reading suggests such a combo is against the rules as written and as intended.