Changing Icon to Client logo icon

the icon you like to insert – is it a colored icon or do you like to use the white globe icon with that green background?

if you like to use it as monochrome icon – i would upload that monochrome globe as svg to fontello and create your custom icon font set.
Then you can use it like all the other entypo-fontello icons.

if you like to use that bicolor globe icon – you can set those icons to display none and insert via pseudo container your colored icon:

#top #main ul.avia-icon-list.av-iconlist-small  li:before {
    content: "" !important;
    width: 40px;
    height: 40px;
    float: left;
    margin-left: -10px;
    margin-right: 10px;
    background-color: transparent;  /**** with background - replace  ***/
    background-image: url(/wp-content/uploads/globe-color.png);
    background-size: 30px 30px;
    background-repeat: no-repeat;
    background-position: center;
    border-radius: 100%;
    top: -3px;
    position: relative;
}

#top .avia-icon-list-container .iconlist_icon {
  display: none 
}

But: to better select your small icon list – it would certainly be advisable to use a custom class.

Yes replacing it at the pseudo-container of the char itself maybe the better option. I do not know where the animation of the list will do the job.
But i guess it will be the char – so try Ismaels code first.

well i can see on your staging site all icons now:

you will know better where your white png is ;)