Like @Shoaib answered, you dont need any jQuery or Javascript. You can to this simply with pure html!
<form method="POST" action="index.php?action=contact_agent">
<select name="agent_id" required>
<option value="1">Agent Homer</option>
<option value="2">Agent Lenny</option>
<option value="3">Agent Carl</option>
</select>
<input type="submit" value="Submit">
</form>
&agent_id=
from form action since you don't need it there.name="agent_id"
to the selectrequired
do indicate that this selection is required.Since you are using PHP, then by posting the form to index.php
you can catch agent_id
with $_POST
/** Since you reference action on `form action` then value of $_GET['action'] will be contact_agent */
$action = $_GET['action'];
/** Value of $_POST['agent_id'] will be selected option value */
$agent_id = $_POST['agent_id'];
As conclusion for such a simple task you should not use any javascript or jQuery. To @FelipeAlvarez that answers your comment