Update package by Windows.Services.Store API on .NET 5.0.

StoreHelper.cs

using System;
using System.Net.NetworkInformation;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Interop;
using Windows.Services.Store;
using WinRT.Interop;

internal static class StoreHelper
{
    /// <summary>
    /// Checks if updated packages are available.
    /// </summary>
    /// <returns>True if available</returns>
    /// <remarks>
    /// If the packages are installed locally or published but as Package flights, this method
    /// will not work correctly.
    /// </remarks>
    public static async Task<bool> CheckUpdateAsync()
    {
        if (!NetworkInterface.GetIsNetworkAvailable())
            return false;

        var context = StoreContext.GetDefault();

        try
        {
            var updates = await context.GetAppAndOptionalStorePackageUpdatesAsync();
            return (updates.Count > 0);
        }
        catch
        {
            return false;
        }
    }

    /// <summary>
    /// Proceeds to download and install updated packages.
    /// </summary>
    /// <param name="window">Owner window</param>
    /// <returns>True if successfully finished downloading and installing</returns>
    public static async Task<bool> ProceedUpdateAsync(Window window)
    {
        if (window is null)
            throw new ArgumentNullException(nameof(window));

        if (!NetworkInterface.GetIsNetworkAvailable())
            return false;

        var context = StoreContext.GetDefault();

        try
        {
            var updates = await context.GetAppAndOptionalStorePackageUpdatesAsync();
            if (updates.Count == 0)
                return false;

            SetOwnerWindow(context, window);

            var result = await context.RequestDownloadAndInstallStorePackageUpdatesAsync(updates);
            return (result.OverallState == StorePackageUpdateState.Completed);
        }
        catch
        {
            return false;
        }
    }

    private static void SetOwnerWindow(StoreContext context, Window window)
    {
        var handle = new WindowInteropHelper(window).Handle;
        InitializeWithWindow.Initialize(context, handle);
    }
}