<input type="hidden" name="admin_email" value="">
в value подставить значение с выбранного selecta<form action="">
<input type="hidden" name="admin_email" value="">
<select name="email">
<option value="option1">option1</option>
<option value="option2">option2</option>
<option value="option3">option3</option>
</select>
<button>Send</button>
</form>
document.querySelector('select[name=email]').addEventListener('change',(e)=>{
const v = e.currentTarget.value
document.querySelector('input[name=admin_email]').value = v
},false)
$('select[name=email]').change(function(){
const v = $(this).val()
$('input[name=admin_email]').val(v)
});