CSS animation triggered through JS only plays every other click

The issue is because the first click adds the class and triggers the animation, yet the second (and every even numbered click) after that removes the class so nothing happens.

To fix this you can use the animationend event to remove the class automatically after the animation has ended. That way when you next click again the class is added to the element once more and the animation plays. Try this:

var anim = document.getElementById("anim");
var willCheck = document.getElementById("contactId");

anim.addEventListener('click', () => {

willCheck.addEventListener('animationend', () => {
#contactId { background: red; }

.highAnim {
  background-color: var(--white);
  animation-name: highcheck;
  animation-duration: 0.35s;
  animation-timing-function: ease-in-out;

@keyframes highcheck {
  0% { transform: rotate(0); }   
  50% { transform: rotate(2deg); }
  100% { transform: rotate(0); }
<a id="anim">Click</a><br><br>
<div id="contactId">Some Text</div>

Note that I removed the 25% and 75% items in the animation as it's a linear motion from 0% to 50% and back again, so they are not needed. This also helps to make the animation smoother.