How to keep :active css style after click a button

CSS

:active denotes the interaction state (so for a button will be applied during press), :focus may be a better choice here. However, the styling will be lost once another element gains focus.

The final potential alternative using CSS would be to use :target, assuming the items being clicked are setting routes (e.g. anchors) within the page- however this can be interrupted if you are using routing (e.g. Angular), however this doesnt seem the case here.

.active:active {
  color: red;
}
.focus:focus {
  color: red;
}
:target {
  color: red;
}

<button class='active'>Active</button>
<button class='focus'>Focus</button>
<a href='#target1' id='target1' class='target'>Target 1</a>
<a href='#target2' id='target2' class='target'>Target 2</a>
<a href='#target3' id='target3' class='target'>Target 3</a>

Javascript / jQuery

As such, there is no way in CSS to absolutely toggle a styled state- if none of the above work for you, you will either need to combine with a change in your HTML (e.g. based on a checkbox) or programatically apply/remove a class using e.g. jQuery

$('button').on('click', function(){
    $('button').removeClass('selected');
    $(this).addClass('selected');
});

button.selected{
  color:red;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>

<button>Item</button><button>Item</button><button>Item</button>
  

I FIGURED IT OUT. SIMPLE, EFFECTIVE NO jQUERY

We're going to to be using a hidden checkbox.
This example includes one "on click - off click 'hover / active' state"

--

To make content itself clickable:

HTML

<input type="checkbox" id="activate-div">
  <label for="activate-div">
   <div class="my-div">
      //MY DIV CONTENT
   </div>
  </label>

CSS

#activate-div{display:none}    

.my-div{background-color:#FFF} 

#activate-div:checked ~ label 
.my-div{background-color:#000}

To make button change content:

HTML

<input type="checkbox" id="activate-div">
   <div class="my-div">
      //MY DIV CONTENT
   </div>

<label for="activate-div">
   //MY BUTTON STUFF
</label>

CSS

#activate-div{display:none}

.my-div{background-color:#FFF} 

#activate-div:checked + 
.my-div{background-color:#000}

Hope it helps!!

In the Divi Theme Documentation, it says that the theme comes with access to 'ePanel' which also has an 'Integration' section.

You should be able to add this code:

<script>
 $( ".et-pb-icon" ).click(function() {
 $( this ).toggleClass( "active" );
 });
</script>

into the the box that says 'Add code to the head of your blog' under the 'Integration' tab, which should get the jQuery working.

Then, you should be able to style your class to what ever you need.