I bumped Lohoris's answer, because its always the right one... sort of. You, as GM, need to be the final arbiter of what is and is not allowable (because someone needs to be). However, it is generally good to discuss things with your players to understand why they made a decision and to work with them to mitigate any worries you might have before disallowing their choice out-right. After all, if we didn't want to be able to do things outside the rules we'd be playing video RPGs instead! But this is a little much for a comment. From the RAW:
Weapons lists the Weapon types this Job is proficient with. Weapon
types not listed in a Job’s profile may not be wielded by a character
unless they have the Advantage Special Training. The one exception
to this are Thrown Weapons, which may be used by all Jobs.
Weapons: Boomerangs, Bows, Crossbows, Rifles
The rules are fairly clear on whether he can USE those weapons. (He took the special training advantage, after all.)
The abilities themselves, on the other hand, don't say anything explicit about weapon usage (other than snipe mentioning a ranged weapon). They do, however use 'fire' repeatedly, which implies a bow, crossbow, or rifle. (Based on the weapons list.)
Still without reading the entire rules, I don't see any particular reason why this combination (although thematically inappropriate to the name Archer) would be particularly problematic. It would simply represent a close combat fighter who excels in hitting critical locations. If you see a particular feature that seems like it would be easily abused (in a way it could not be if using a ranged weapon) then you might want to discuss limiting that ability.