How can I get StackPanel to use an ItemTemplate?
ItemsControl is essentially a StackPanel with an ItemTemplate. It uses a StackPanel internally.
However, it looks like you're trying to display a single customer rather than a list of them (I sound like Clippy, don't I?). In that case you want to use a ContentControl:
<ContentControl Content="{Binding SelectedCustomer}" ContentTemplate="{StaticResource CustomerTemplate}" />