Ref:

Change (Modify) connection string in Web.Config at Runtime in ASP.Net using C# and VB.Net

Add the App.config file in your project.

In that just add this line

<?xml version="1.0" encoding="utf-8"?>
<configuration>
  <connectionStrings>
  </connectionStrings>
</configuration>

In the Form add a TextBox to name the Connection String. On clicking of button connectionString is added in App.Config file.

Form

Namespaces

using System.Windows.Forms;
using System.Xml;
using System.Data.SqlClient;
using System.IO;

C#

public void add(string name)
{
    string ApplicationPath = Application.StartupPath;
    string YourPath = Path.GetDirectoryName(ApplicationPath);
    bool isNew = false;

    string path = Path.GetDirectoryName(YourPath) + "\\App.config";
    XmlDocument doc = new XmlDocument();
    doc.Load(path);
    XmlNodeList list = doc.DocumentElement.SelectNodes(string.Format("connectionStrings/add[@name='{0}']", name));
    XmlNode node;
    isNew = list.Count == 0;
    if (isNew)
    {
        node = doc.CreateNode(XmlNodeType.Element, "add", null);
        XmlAttribute attribute = doc.CreateAttribute("name");
        attribute.Value = name;
        node.Attributes.Append(attribute);

        attribute = doc.CreateAttribute("connectionString");
        attribute.Value = "";
        node.Attributes.Append(attribute);

        attribute = doc.CreateAttribute("providerName");
        attribute.Value = "System.Data.SqlClient";
        node.Attributes.Append(attribute);
    }
    else
    {
        node = list[0];
    }
    string conString = node.Attributes["connectionString"].Value;
    SqlConnectionStringBuilder conStringBuilder = new SqlConnectionStringBuilder(conString);
    conStringBuilder.InitialCatalog = "Sample";
    conStringBuilder.DataSource = "myserver";
    conStringBuilder.IntegratedSecurity = false;
    conStringBuilder.UserID = "test";
    conStringBuilder.Password = "12345";
    node.Attributes["connectionString"].Value = conStringBuilder.ConnectionString;
    if (isNew)
    {
        doc.DocumentElement.SelectNodes("connectionStrings")[0].AppendChild(node);
    }
    doc.Save(path);
}

private void button1_Click(object sender, EventArgs e)
{
    if (!string.IsNullOrEmpty(this.textBox1.Text.Trim()))
    {
        add(this.textBox1.Text.Trim());
    }
}

VB

Namespaces

Imports System.IO
Imports System.Xml
Imports System.Data.SqlClient

VB Code

Private Sub button1_Click(sender As System.Object, e As System.EventArgs) Handles button1.Click
    If Not String.IsNullOrEmpty(Me.textBox1.Text.Trim()) Then
        add(Me.textBox1.Text.Trim())
    End If
End Sub

Public Sub add(name As String)
    Dim ApplicationPath As String = Application.StartupPath
    Dim YourPath As String = Path.GetDirectoryName(ApplicationPath)
    Dim isNew As Boolean = False

    Dim path__1 As String = Path.GetDirectoryName(YourPath) + "\app1.config"
    Dim doc As New XmlDocument()
    doc.Load(path__1)
    Dim list As XmlNodeList = doc.DocumentElement.SelectNodes(String.Format("connectionStrings/add[@name='{0}']", name))
    Dim node As XmlNode
    isNew = list.Count = 0
    If isNew Then
        node = doc.CreateNode(XmlNodeType.Element, "add", Nothing)
        Dim attribute As XmlAttribute = doc.CreateAttribute("name")
        attribute.Value = name
        node.Attributes.Append(attribute)

        attribute = doc.CreateAttribute("connectionString")
        attribute.Value = ""
        node.Attributes.Append(attribute)

        attribute = doc.CreateAttribute("providerName")
        attribute.Value = "System.Data.SqlClient"
        node.Attributes.Append(attribute)
    Else
        node = list(0)
    End If
    Dim conString As String = node.Attributes("connectionString").Value
    Dim conStringBuilder As New SqlConnectionStringBuilder(conString)
    conStringBuilder.InitialCatalog = "Sample"
    conStringBuilder.DataSource = "myserver"
    conStringBuilder.IntegratedSecurity = False
    conStringBuilder.UserID = "test"
    conStringBuilder.Password = "12345"
    node.Attributes("connectionString").Value = conStringBuilder.ConnectionString
    If isNew Then
        doc.DocumentElement.SelectNodes("connectionStrings")(0).AppendChild(node)
    End If
    doc.Save(path__1)
End Sub