ASP.NET C# Dropdown List using User Control

The first problem I see is that you are inheriting from Page inside your UserControl:

public partial class _Default : System.Web.UI.Page

Inherit from UserControl instead.

// notice that I also renamed the class to match the control name
public partial class Vendor : System.Web.UI.UserControl

Using a Codebehind File

As @x0n pointed out, your user control code can be placed in a codebehind file (automatically created when you create a user control inside Visual Studio). User controls typically consist of a markup portion (.ascx), a codebehind (.ascx.cs), and a designer file (.ascx.designer.cs). HTML markup goes into the ASCX file, and binding code goes into the codebehind.

I'd suggest saving your code, deleting your current user control, and re-adding it through Visual Studio.

Sample Project Structure

Markup (ASCX) File

<%@ Control Language="C#" AutoEventWireup="true" CodeBehind="VendorListControl.ascx.cs" Inherits="MyNamespace.VendorListControl" %>
<asp:DropDownList runat="server" ID="ddlVendorList" />
<asp:Label runat="server" ID="lblMessage" />

Codebehind

using System;
using System.Configuration;
using System.Data;
using System.Data.SqlClient;

namespace MyNamespace
{
    public partial class VendorListControl : System.Web.UI.UserControl
    {
        protected void Page_Load( object sender, EventArgs e ) {
            if( !IsPostBack ) {
                FillVendors();
            }
        }

        private void FillVendors() {
            string strConn = ConfigurationManager.ConnectionStrings["ConnectionString"].ConnectionString;
            System.Data.SqlClient.SqlConnection conn = new System.Data.SqlClient.SqlConnection( strConn );

            SqlCommand cmd = new SqlCommand();
            cmd.Connection = conn;
            cmd.CommandType = CommandType.Text;
            cmd.CommandText = "SELECT VendorID, VendorName FROM Vendor";

            DataSet objDs = new DataSet();
            SqlDataAdapter dAdapter = new SqlDataAdapter();
            dAdapter.SelectCommand = cmd; ;
            conn.Open();
            dAdapter.Fill( objDs );
            conn.Close();

            if( objDs.Tables[0].Rows.Count > 0 ) {
                this.ddlVendorList.DataSource = objDs.Tables[0];
                this.ddlVendorList.DataTextField = "VendorName";
                this.ddlVendorList.DataValueField = "VendorID";
                this.ddlVendorList.DataBind();
                this.ddlVendorList.Items.Insert( 0, "-- Select --" );
            }
            else {
                this.lblMessage.Text = "No Vendor Found";
            }
        }
    }
}

Alternate Method - Remove Class Declaration

If you don't want to add a codebehind file for some reason, remove the class declaration altogether and just include the code inside it.

<script runat="server">
    protected void Page_Load(object sender, EventArgs e){
        if (!IsPostBack){
            FillVendor();
        }
    }

    // etc
</script>

As a side note, I would put the data access logic in a separate class for proper separation/reuse, but the structure you've outlined should work once you've corrected the aforementioned problems.