You can do it with javascript
<script>
var a = document.getElementById('tfnewsearch');
a.addEventListener('submit',function(e) {
e.preventDefault();
var b = document.getElementById('tftextinput').value;
window.location.href = 'http://mywebsite.com/'+b;
});
</script>
and give your input field an ID called tftextinput.
<input type="text" class="tftextinput" id="tftextinput" name="q" size="21" maxlength="120">
I don't really understand why you would like to do this, as it is way more difficult to handle this request server side as it would be to simply handle the $_GET data you will receive when doing a standard form transmission.
EDIT:
Here is the full code:
<div id="tfheader">
<form id="tfnewsearch" method="get" action="http://www.mywebsite.com">
<input type="text" class="tftextinput" id="tftextinput" name="q" size="21" maxlength="120"><input type="submit" value="search" class="tfbutton">
</form>
<div class="tfclear"></div>
</div>
<script>
var a = document.getElementById('tfnewsearch');
a.addEventListener('submit',function(e) {
e.preventDefault();
var b = document.getElementById('tftextinput').value;
window.location.href = 'http://mywebsite.com/'+b;
});
</script>