Just remove the submit button from templates/layout.tpl.html
<label for="ajaxSearch_submit">
<input id="ajaxSearch_submit" type="submit" name="sub" value="[+as.submitText+]" />
</label>
Or even better:
In snippet.ajaxSearch.php, add this:
// $AS_showSubmit [0 | 1]
// Show or hide the submit button (1=true, 0=false)
$cfg['AS_showSubmit'] = isset($AS_showSubmit ) ? $AS_showSubmit : (isset($__AS_showSubmit ) ? $__AS_showSubmit : 1);
Then you can add this to layout.tpl.html:
[+as.showSubmit:is=`1`:then=`
<label for="ajaxSearch_submit">
<input id="ajaxSearch_submit" type="submit" name="sub" value="[+as.submitText+]" />
</label>
`+]
and add this param to your existing AjaxSearch snippet call: &AS_showSubmit=`0`
Or define it in the config file, i.e. create a global var.
see configs/default.config.php
(all of this is untested, btw)