You can use the edit event for the grid and change the text box to a dropdown list in that event.
So
columns.Template(o => @Html.DropDownList("Department_Name", new SelectList(ViewBag.Department_Name))).Width(100).Column.Title = "Dept Name";
would change to this.
columns.Bound(d => d.Department_Name).Width(100).Title("Dept Name");
Than you would create your OnEdit event to change the text box to a dropdown and populate the dropdown with its options.
<script type="text/javascript">
function Grid_onEdit(e) {
var dataItem = e.dataItem;
var mode = e.mode;
var form = e.form;
if (mode == "edit") {
var parent = document.getElementById(e.currentTarget.id)
var deptTextBox = $(form).find("#Department_Name");
var deptSelect = document.createElement('select');
deptSelect.Id = "Department_Name";
deptSelect.Name = "Departmant_Name";
foreach (var dept in departments)
{
deptSelect.add(dept);
}
parent.replace(deptTextBox.get(0), deptSelect);
}
}
</script>
This code is just an example and probably won't work right out of the box but should get you started.