You can just use a form as below, with input type submit, which in this case, if you press enter in any input - if you had more of them - it will be a default behaviour of the form to be submitted:
<form id="search">
<input type='text' id='searchText' />
<input type='submit' value='Search' />
</form>
or, as it shows, you want to use the onsubmit function and handle the "submit" of the form, so you can do this:
<form id="search" action="#">
<input type="text" id='searchText' name="myinput" onkeypress="handle" />
</form>
<script>
function handle(e){
if(e.key === "Enter"){
alert("Enter was just pressed.");
}
return false;
}
</script>
A code, quite the same, can be found on this similar question: How to capture Enter key press?
Hope I answered your question, even out of time.