If a PC completes their move climbing out of a pit, is it reasonable to allow them to use their bow as an Attack?
I.e. to clamber out, take the bow off their back, find an arrow, notch the arrow, target a foe and unleash it in 1 turn.
The section of the rules on Your Turn read:
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. Your speed--sometimes called your walking speed--is noted on your character sheet.
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
Drawing the ammunition from a quiver, case, or other container is part of the attack
Rules as written, the character would not only be able to do that, but also take a bonus action.
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.