If a script is using document.write() to emit content, then you have to place the script in the place that you want the content to go.
If you control the script, then you can have the script not emit it's content by default and you can call a function in the script and pass a parent object that directs the script where it should put it's content (which it will not use document.write() to create).
The only other option is to place a div around the script, let it emit the content into that div and then move that container div to another location in the page after it runs.
<div id="ads">
<script type="text/javascript" src="http://website.com/wp-content/plugins/oiopub-direct/js.php?type=banner&align=center&zone=1"></script>
</div>
<script>
$(document).ready(function() {
$("#ads").appendTo("#myFooter");
});
</script>
I'm not sure exactly what you're trying to ask about jQuery. jQuery is capable of inserting dynamically created content anywhere you want in the page. You'd have to ask a more specific question for us to advise in more details.