First check if the vibration is not mechanical.

Second and easiest to do, I would swap the X and Y motor plugs from the drives, the 6 wire connector, assuming two motors are identical and powered by the same psu. Now when you jog the X axis, the Y will be moving and vice versa. If the other axis is vibrating then the drive is faulty. NEVER unplug the motor and power plug from the drives before turning off the power!!
I cannot see if there is a plug at the motor end. If it has a plug, you could easily swap the motors as John said and this way you find if it's the stepper, cable or drive faulty.

Third but most likely, I would check/replace the motor cable, if your Y axis cable is the one that is moving inside a cable chain.

The stepper motor is the last thing I would suspect to fail.