Do modern aircraft require rudder input in order to perform a coordinated turn?
Most transport aircraft use yaw damper systems to take care of minor rudder inputs.  Most autopilot systems are actually only 2-axis - pitch and roll since the rudder's job is only to keep the tail lined up behind the nose.  The yaw damper is a separate "autopilot" system and has limited authority, sufficient to deal with minor yaw disturbances, dampen dutch roll, and counter adverse yaw from the ailerons and is active all the time whether the autopilot is on or not.

With a Y/D system the only time a pilot really needs to make a rudder input in flight is if an engine quits, because the yaw damper's authority is not sufficient to counter asymmetric thrust.  Generally in a jet, once through the departure profile, feet are on the floor even when hand flying.  They are only on the pedals to steer the nosewheel during the takeoff and be ready in case an engine quits.

On a swept wing aircraft the yaw damper is essential; if the yaw damper is off and yaw disturbances occur, dutch roll motions can start and if the pilot tries to respond with rudder inputs he almost always can't stay in phase and things get exciting.  Because of this criticality the Y/D is usually dual channel.  One of the tests done on production aircraft is to induce a large yaw movement with both YD channels off to get a dutch roll going, then engage each YD channel and make sure it stops the dutch roll.