Is it legal to shoot carp with a bow?

Yes, carp may be taken with a bow. 

All public waters are open to bow fishing, unless posted otherwise, so consult local rules before bow fishing.  Bow fishermen must have in possession a valid Kansas fishing license, unless exempt by law.  Arrows must have barbed heads, and each arrow must be attached by a line to the bow and must be shot from the bow.  Waters within 50 yards of an occupied boat dock or ramp, occupied swimming area, occupied picnic site or camping area, and other public use areas are closed to bow fishing.  Firearms may not be used to take fish.