Create a method call in .NET based on a string value

You've got a deeper problem. Your strings are too important. Who is passing you strings? can you make them not do that?

Stick with the switch statement, as it decouples your internal implementation (method names) from your external view.

Suppose you localize this to German. You gonna rename all those methods?

Type type = GetType();
MethodInfo method = type.GetMethod("Show"+reportName+"Report");
if (method != null)
{
    method.Invoke(this, null);
}

This is C#, should be easy enough to turn it into VB. If you need to pass parameter into the method, they can be added in the 2nd argument to Invoke.