As we know that the zendesk Messaging Widget doesn't have proactive chat natively, after deep research I made this script to help anyone who wants the widget to pop a while after the website loads.
Hope it helps you!
<script type="text/javascript">
window.setTimeout(function() {
zE(function(){
zE('messenger', 'open');
});
//You can add more APIs on this line
}, 10000); //time’s in milliseconds - 10 seconds = 100000
</script>