How can I sort a ListBox using only XAML and no code-behind?

Use a CollectionViewSource:

<CollectionViewSource x:Key="SortedItems" Source="{Binding CollectionOfStrings}"
    xmlns:scm="clr-namespace:System.ComponentModel;assembly=Win‌​dowsBase">
    <CollectionViewSource.SortDescriptions>
        <scm:SortDescription PropertyName="SomePropertyOnYourItems"/>
    </CollectionViewSource.SortDescriptions>
</CollectionViewSource>

<ListBox ItemsSource="{Binding Source={StaticResource SortedItems}}"/>

You might want to wrap your strings in a custom VM class so you can more easily apply sorting behavior.