How do I access Window Resources in WinUI 3

You can declare the resource in the root control of your window.

<Window ...>
    <Border ...>
        <Border.Resources>
            <Style TargetType="Button">
                <Setter Property="Margin" Value="10"/>
            </Style>
        </Border.Resources>
        <Button ... />
    </Border>
</Window>

Apparently it was a design choice, to quote Miguel, a member of Microsoft, from a Github issue:

[...] the Window object doesn't have Resource property like WPF, for instance. It was a design decision [...]

The alternative is to use the dictionary within the component's context, for instance:

<ListView>
  <ListView.Resources>
   <!-- My resources -->
  </ListView.Resources>
</ListView>

You can't add resources to a Window. But you can add resources like below depending on the scope you need those resources.

App project (App.xaml)

<Application
    x:Class="WinUIDemoApp.App"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:local="using:WinUIDemoApp">
    <Application.Resources>
        <ResourceDictionary>
            <ResourceDictionary.MergedDictionaries>
                <XamlControlsResources xmlns="using:Microsoft.UI.Xaml.Controls" />
                <!--  Other merged dictionaries here  -->
            </ResourceDictionary.MergedDictionaries>
            <!--  Other app resources here  -->
            <SolidColorBrush x:Key="AppProjectScopeBrush" Color="SkyBlue" />
        </ResourceDictionary>
    </Application.Resources>
</Application>

Page/Control

<Page
    x:Class="WinUIDemoApp.SomePage"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
    xmlns:local="using:WinUIDemoApp"
    xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
    Background="{ThemeResource ApplicationPageBackgroundThemeBrush}"
    mc:Ignorable="d">

    <Page.Resources>
        <SolidColorBrush x:Key="SomePageScopeBrush" Color="SkyBlue" />
    </Page.Resources>

    <Grid RowDefinitions="Auto,*">
        <StackPanel Grid.Row="0">
            <Button Background="{StaticResource SomePageScopeBrush}" />
            <!--
                DOES NOT WORK: Can not find the ListViewScopeBrush.
                <Button Background="{StaticResource ListViewScopeBrush}" />
            -->
        </StackPanel>
        <ListView Grid.Row="1">
            <ListView.Resources>
                <SolidColorBrush x:Key="ListViewScopeBrush" Color="SkyBlue" />
            </ListView.Resources>
            <ListView.ItemTemplate>
                <DataTemplate>
                    <Button
                        Foreground="{StaticResource SomePageScopeBrush}"
                        Background="{StaticResource ListViewScopeBrush}" />
                </DataTemplate>
            </ListView.ItemTemplate>
        </ListView>
    </Grid>

</Page>

You're right - there are no Window resources. You can have globally shared resources by making App resources, or you can have <page> and <Control> level resources.

What your probably want is to define your resource dictionaries in App.xml

<Application
x:Class="BOMCheck.App"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:converters="using:CommunityToolkit.WinUI.UI.Converters"
xmlns:helpers="using:BOMCheck.Helpers"
xmlns:local="using:BOMCheck"
xmlns:media="using:Microsoft.UI.Xaml.Media"
RequestedTheme="Light">
<Application.Resources>
    <ResourceDictionary>
        <ResourceDictionary.MergedDictionaries>