The question is not very clear but assuming the page www.mywebsite.com/somesearch/searchtext is resolving to a page that can access the query string in the URL and process the search with the searchtext you can do something like this:
<html>
<head>
<script>
$('#searchButton').click(function(){
var searchText = document.getElementById('searchBox').value;
window.location.href = 'http://www.mywebsite/somesearch/'+encodeURIComponent(searchText);
})
</script>
</head>
<body>
<input type='text' id='searchBox' />
<input type='submit' id='searchButton' value='Search' />
</body>
</html>
You can use one of the available modules that let you write JavaScript, HTML or PHP code as a module listed here:
http://extensions.joomla.org/extensions/edition/custom-code-in-modules
If you do this you can skip the html, head and body tags. Just put the <script>...</script>
and then the input
.
Also be careful processing the user input directly without escaping it against possible sql injection attacks.