You are correct to say that the acceleration during the jump will not be constant. However, once the jumper is in the air, only gravity (ignoring air resistance and other effects) acts to accelerate the body. The details of the jump are not important to applying kinematic equations to the motion, so long as you know the initial velocity (angle and magnitude).
Also, note that this assumes you're working with a point mass, i.e. no rotation or internal energy, which isn't strictly accurate for the real world. Insofar is you can approximate things to this ideal case, the kinematic equations will track the center of mass of the jumper. Of course, real long jumpers rotate their bodies so their feet land farther from the starting point than their center of mass, but if you're only concerned with where the center of mass is, not the jump distance that would be measured in a competition, kinematics will serve you acceptably well.