required field valodator is used to check where a browser
has entered a value in atextbox or not.

Hi,
We can add the code in page_load

If Me.RadioButton1.Checked = True Then
            Button1.CausesValidation = True
        Else
            If Me.RadioButton2.Checked = True Then
                Me.Button1.CausesValidation = False
            End If
        End If

Required Field Validator does not depend upon any server 
control either it is enable or disable.Means if radio 
button1 is enable and radiobutton2 is disable at same time 
then u press button then by default validation control 
fired also in vice-versa case

Set causes validation on the button to false. Add a custom 
javascript function to the onclick event of that button. In 
the javascript function check whether the radio button is 
checked or not then. If checked call ValidatorEnable
(regValidatorId, true) and then call Page_ClientValidate() 
function.

Required validator checks for TextBox empty or not..

so to solve this problem

RequiredField validator--->properties..>validationGroup:sbmt
Button                 --->properties..>validationGroup:sbmt

Set both radio buttons AutoPostBack property to true.

and following code in page_load:

 if (RadioButtonNo.Checked == true)
        {
            this.Button1.CausesValidation = false;
        }
        else
        {
            this.Button1.CausesValidation = true;
        }