Random in CSS or JS

You can do this with css custom properties. The first thing to do is create a property for your transform:

:root{
  --rot: rotate(108000deg);
}

And then use that in place of your hard-coded value

@keyframes wheel {
  from { transform: rotate(0);}             
  to { transform: var(--rot); }
}

At this stage everything should carry on working as before.

Now you can manipulate that property with javascipt:

var min = 360;
var max = 108000;
var rnd = Math.random()* (max - min) + min;
console.log(rnd);

var wheel = document.querySelector(".wheel");
wheel.style.setProperty("--rot","rotate(" + rnd + "deg)");
:root{
  --rot: rotate(108000deg);
}

.wheel {				
  animation: wheel 3s .5s;	
  animation-fill-mode: both;	
}

@keyframes wheel {
  from { transform: rotate(0);}				
  to { transform: var(--rot); }
}
<head>
<link rel="stylesheet" href="bottle.css">
<script src="bottle.js"></script>
</head>
<img id="wheel" class="wheel" src="https://cdn.iconscout.com/icon/free/png-256/grinning-face-smile-emoji-happy-37705.png">

If you want to repeat the animation on something like a click you need to doa bit of a workaround

var min = 360;
var max = 108000;

var wheel = document.querySelector("#wheel");
wheel.addEventListener("click", function(e) {
  e.preventDefault;
  
  // -> removing the class
  wheel.classList.remove("wheel");
  
  // -> triggering reflow /* The actual magic */
  // without this it wouldn't work. Try uncommenting the line and the transition won't be retriggered.
  // Oops! This won't work in strict mode. Thanks Felis Phasma!
  // element.offsetWidth = element.offsetWidth;
  // Do this instead:
  void wheel.offsetWidth;
  
  var rnd = Math.random()* (max - min) + min;
  console.log(rnd);
  wheel.style.setProperty("--rot","rotate(" + rnd + "deg)");
  // -> and re-adding the class
  wheel.classList.add("wheel");
}, false);
:root{
  --rot: rotate(108000deg);
}

.wheel {				
  animation: wheel 3s .5s;	
  animation-fill-mode: both;	
}

@keyframes wheel {			
  to { transform: var(--rot); }
}
<head>
<link rel="stylesheet" href="bottle.css">
<script src="bottle.js"></script>
</head>
<img id="wheel" src="https://cdn.iconscout.com/icon/free/png-256/grinning-face-smile-emoji-happy-37705.png">