<body>
<select class="spisok" onchange="selRadio(this.value)">
<option value="0" class="no_select">Выберите ваш район</option>
<option value="1">Район1</option>
<option value="2">Район2</option>
</select>
<input type="radio" name="delivery" value="1" class="delivery-item" id="del-type-1">
<input type="radio" name="delivery" value="2" class="delivery-item" id="del-type-2">
<script>
function selRadio(val) {
var rad = document.getElementsByName("delivery");
for (var i = 0; i < rad.length; i++) {
rad[i].checked = rad[i].id == "del-type-" + val;
}
}
</script>
</body>
можно так и еще есть куча вариантов