I would use a string builder object to achieve this:
Dim s As New StringBuilder
Dim final As String
For Each row As GridViewRow In gvmstrgrid.Rows
Dim chk As CheckBox = DirectCast(row.FindControl("chkbx1"), CheckBox)
If chk.Checked = True Then
Dim str1 As String
str1 = DirectCast(row.FindControl("lblamc"), Label).Text
s.Append(str1)
End If
Next
final = s.ToString()
Something like that I believe will do what you are after.
Please note where I moved your declarations of the string builder and the final string to.