<div class="zakaz-form switch-form">Содержимое 1-го блока + 1-я кнопка</div>
<div class="cons-form switch-form">Содержимое 2-го блока + 2-я кнопка</div>

.zakaz-form{
  animation: switch-zakaz forwards 3s;
  -webkit-animation-play-state:paused;
  animation-play-state:paused;
}
@keyframes switch-zakaz {
  0% {
    opacity: 1;
    left: 0;
  }
  20% {
    opacity: 1;
    left: 0;
  }
  100% {
    opacity: 0;
    left: 95%;
  }
}

.cons-form{
  animation: switch-cons forwards 3s;
  -webkit-animation-play-state:paused;
  animation-play-state:paused;
}
@keyframes switch-cons {
  0% {
    opacity: 0;
    left: -1000px;
  }
  80% {
    opacity: 1;
    left: calc(50% - 275px);
  }
  100% {
    opacity: 1;
    left: calc(50% - 275px);
  }
}

.start-anim {
    -webkit-animation-play-state:running;
    animation-play-state:running;
}

jQuery(document).ready(function($){
        $(function(){
            $('.cons-button').click(function(){
                $('.switch-form').toggleClass('start-anim');
            });
        });
});