更新

您不必在xaml中为选项卡式页面设置ViewModel。把它放在.cs文件中就足够了。你可以试试下面的代码：
对于选项卡页面：

<NavigationPage Title="Cart" IconImageSource="cart.png">
    <x:Arguments>
        <local:CartPage Title="Cart">
        </local:CartPage>
    </x:Arguments>
</NavigationPage>

在CartPage.cs中

public partial class CartPage : ContentPage
{
    CartViewModel ViewModel;
    public CartPage()
    {
        InitializeComponent();
        ViewModel = new CartViewModel();
        BindingContext = ViewModel;

    }

更新
1.要使用RadioButton，您可以尝试以下方法：

In .xaml

<RadioButton BorderColor="Red" Content="Collect" Value="Collect" CheckedChanged="RadioButton_CheckedChanged"/>
<RadioButton x:Name="Delivery" Content="Delivery" CheckedChanged="RadioButton_CheckedChanged"/>

在.cs文件中

public partial class CartPage : ContentPage
{
    CartViewModel ViewModel;
    public CartPage(CartViewModel viewModels)
    {
        InitializeComponent();
        ViewModel = viewModels;
        BindingContext = ViewModel;

    }

    async void RadioButton_CheckedChanged(System.Object sender, Xamarin.Forms.CheckedChangedEventArgs e)
    {
        RadioButton r = sender as RadioButton;
        if((string)(r.Content) == "Delivery" && r.IsChecked)
        {
            viewModel.TotalOrder += 50;

        }
        if((string)(r.Content) == "Collect" && r.IsChecked)
        {
            await DisplayAlert("Alert", "You have been alerted", "OK");

        }
    }
}

2.改变颜色

如果使用Android，则可以使用Custom Renderer
1.在共享项目中创建自己的自定义RadioButton：

public class MyRadioButton : RadioButton
{
    public MyRadioButton()
    {
    }
}

2.然后在YourProject.Android文件夹中创建MyRadioButtonRenderer.cs

[assembly: ExportRenderer(typeof(MyRadioButton), typeof(MyRadioButtonRenderer))]
namespace ViewPageLabelNotRefreshing76512008.Droid
{
    public class MyRadioButtonRenderer : RadioButtonRenderer
    {
        public MyRadioButtonRenderer(Context context) : base(context)
        {
        }

        protected override void OnElementChanged(ElementChangedEventArgs<RadioButton> e)
        {
            base.OnElementChanged(e);

            if (Control != null)
            {
                Control.ButtonTintList = ColorStateList.ValueOf(Android.Graphics.Color.ForestGreen);
            }
        }
    }
}

3.在XAML中使用它

<ContentPage
...
    xmlns:local="clr-namespace:ViewPageLabelRefreshing">

<local:MyRadioButton x:Name="Delivery" Content="Delivery" CheckedChanged="RadioButton_CheckedChanged"/>

===============================
我根据你的代码做了一些修改。你的项目很长，我只关注了这个问题删除命令部分。我没有对paycommand部分进行更改。所以在xaml中，我只是注解掉了以下部分：

<!--<RadioButton Content="Collect" Value="Collect" CheckedChanged="RadioButton_CheckedChanged"/>
    <RadioButton x:Name="Delivery" Content=""/>
    <Button Text="Fulfill Your Pets Wishlist"
            CornerRadius="20"
            BackgroundColor="Transparent"
            BorderColor="ForestGreen"
            BorderWidth="1.5"
            Clicked="Button_Clicked"
            />-->

我让CartPage.cs文件更容易（因为我们使用了mvvm方式）：

public partial class CartPage : ContentPage
{
    public CartPage()
    {
        InitializeComponent();
        this.BindingContext = new CartViewModel();
    }
}

然后在CartViewModel中

public class CartViewModel : INotifyPropertyChanged
{
    //define some variable or constant. No need to show them here
    ...

    //I add these two properties and implement OnPropertyChanged
    private string _strPoints;
    public string strPoints
    {
        get
        {
            return _strPoints;
        }
        set
        {
            _strPoints = value;
            OnPropertyChanged(nameof(strPoints));
        }
    }

    private string _strTotalOrder;
    public string strTotalOrder
    {
        get
        {
            return _strTotalOrder;
        }
        set
        {
            _strTotalOrder = value;
            OnPropertyChanged(nameof(strTotalOrder));
        }
    }

    //for  Carts you can just define like this
    public ObservableCollection<Cart> Carts { get; set; } = new ObservableCollection<Cart>();

    public CartViewModel()
    {
        //instead of execute in page.cs OnAppearing, you can just
        //execute the command in constructor
        LoadCartsCommand.Execute(null);
        LoadTotalsCommand.Execute(null);
    }

    private ICommand _loadCartsCommand;
    public ICommand LoadCartsCommand
    {
        get
        {
            if (_loadCartsCommand == null)
            {
                _loadCartsCommand = new Command(() => LoadCarts());
            }

            return _loadCartsCommand;
        }
    }

    public void LoadCarts()
    {
        // Call the GetCategories method to load the categories from the database
        GetCarts();
    }

    private ICommand _loadTotalsCommand;
    public ICommand LoadTotalsCommand
    {
        get
        {
            if (_loadTotalsCommand == null)
            {
                _loadTotalsCommand = new Command(() => LoadTotals());
                OnPropertyChanged();
            }

            return _loadTotalsCommand;
        }
    }

    // Method for getting the categories from the database
    private void GetCarts()
    {
        //In this method, you could add each Cart instance into Carts collection just as your code. In my demo, I just add some test data. Also to reduce code not show here.
        ...
    }

    public void LoadTotals()
    {
        //You make some calculation here, just as your code
        ...
    }

    private ICommand _cartCommand;

    public ICommand CartsCommand
    {
        get
        {
            if (_cartCommand == null)
            {
                _cartCommand = new Command<Cart>(async (cart) =>
                {
                    await OnCartSelected(cart);
                });
            }
            return _cartCommand;
        }
    }

    public async Task OnCartSelected(Cart cart)
    {
        // You should do these things
        // 1.First you should remove the this deleted cart in Carts collection.(The UI also refresh automatically) you could simply use the following: 
        Carts.Remove(cart);

        //2.Delete the cart in your mysql.  shouldn't it be DELETE FROM cart WHERE ordernumber =  strOrderNumber4 and name = cart.Name?
        //3. recalculation as the mysql has changed
        LoadTotalsCommand.Execute(null);
    }

......
}

希望能帮上忙！