1. Login to your App Console in Twitter.
2. Add Privacy Policy and Terms Conditions Links in Settings Tab.
3. Then in Permissions Tab check the Request email addresses CheckBox and click Update Settings
4. Download the Latest ASPSnippets Twitter API DLL using the following link.
Download ASPSnippets Twitter API DLL
5. Finally use the following article and just add one more line to get the Email.
Add Twitter Login Authentication to website in ASP.Net using C# and VB.Net
C#
lblEmail.Text = dt.Rows[0]["email"].ToString();
VB.Net
lblEmail.Text = dt.Rows(0)("email").ToString()