You need to use some AJAX for this. Or well you could pass a iframe by doing...
<form name="search" action="javascript:void();" method="get">
Enter Search Term: <input type="text" id="keyword" name="search">
<input type="submit" onClick="search()" value="Submit">
</form>
<script>
function search() {
var searchIframe = document.createElement("iframe");
searchIframe.setAttribute("src","search.php?search="+document.getElementById("keyword").value);
searchIframe.setAttribute("className","Inset your class with CSS props");
document.getElementById("shadowBox").appendChild(searchIframe); //Put it in your shadowBox so it will display in the right position you want.
}
</script>
That should get it for you. It should implement your iframe for you.