
HammerJack |
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Climbing requires two hands normally.
Climbing with a Climbing Speed requires the same number of hands.
Climbing with the Combat Climber feat allows you to have 1 hand occupied.
Climbing up a wall and hanging out up there while using both hands for a bow calls for the Cliffside Lizardfolk heritage, or possibly Slippers of Spuder Climb.
For most characters, unless there was some outcropping they could climb to and stand on, the way it works is "not at all".

graystone |

Combat Climber allows a single hand to be used in combat so any 1 handed weapon is on the table. So a repeating hand crossbow is a good option if you want to shot and climb.
If you're a Tailed Goblin, you reduce the number of hands needed by one and since they get Combat Climber as a bonus feat that they would be able to fire a bow while climbing. Caveclimber Kobold, Vine Leshy and Cliffscale Lizardfolk can use their feet, tail and/or vines instead of their hands so they too can fire a bow while climbing.
EDIT: Cloak of the Bat allows you to "use your feet to hang from any surface that can support your weight" so you might also be able to use it to fire a bow but it would require drawing the bow once you stop climbing.