WPF Data Triggers and Story Boards

What you want is possible by declaring the animation on the progressWheel itself:
The XAML:

<UserControl x:Class="TriggerSpike.UserControl1"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Height="300" Width="300">
<UserControl.Resources>
    <DoubleAnimation x:Key="SearchAnimation" Storyboard.TargetProperty="Opacity" To="1" Duration="0:0:4"/>
    <DoubleAnimation x:Key="StopSearchAnimation" Storyboard.TargetProperty="Opacity" To="0" Duration="0:0:4"/>
</UserControl.Resources>
<StackPanel>
    <TextBlock Name="progressWheel" TextAlignment="Center" Opacity="0">
        <TextBlock.Style>
            <Style>
                <Style.Triggers>
                    <DataTrigger Binding="{Binding IsBusy}" Value="True">
                        <DataTrigger.EnterActions>
                            <BeginStoryboard>
                                <Storyboard>
                                    <StaticResource ResourceKey="SearchAnimation"/>
                                </Storyboard>
                            </BeginStoryboard>
                        </DataTrigger.EnterActions>
                        <DataTrigger.ExitActions>
                            <BeginStoryboard>
                                <Storyboard>
                                   <StaticResource ResourceKey="StopSearchAnimation"/> 
                                </Storyboard>
                            </BeginStoryboard>
                        </DataTrigger.ExitActions>
                    </DataTrigger>
                </Style.Triggers>
            </Style>
        </TextBlock.Style>
        Searching
    </TextBlock>
    <Label Content="Here your search query"/>
    <TextBox Text="{Binding SearchClause}"/>
    <Button Click="Button_Click">Search!</Button>
    <TextBlock Text="{Binding Result}"/>
</StackPanel>

Code behind:

    using System.Windows;
using System.Windows.Controls;

namespace TriggerSpike
{
    public partial class UserControl1 : UserControl
    {
        private MyViewModel myModel;

        public UserControl1()
        {
            myModel=new MyViewModel();
            DataContext = myModel;
            InitializeComponent();
        }

        private void Button_Click(object sender, RoutedEventArgs e)
        {
            myModel.Search(myModel.SearchClause);
        }
    }
}

The viewmodel:

    using System.ComponentModel;
using System.Threading;
using System.Windows;

namespace TriggerSpike
{
    class MyViewModel:DependencyObject
    {

        public string SearchClause{ get;set;}

        public bool IsBusy
        {
            get { return (bool)GetValue(IsBusyProperty); }
            set { SetValue(IsBusyProperty, value); }
        }

        public static readonly DependencyProperty IsBusyProperty =
            DependencyProperty.Register("IsBusy", typeof(bool), typeof(MyViewModel), new UIPropertyMetadata(false));



        public string Result
        {
            get { return (string)GetValue(ResultProperty); }
            set { SetValue(ResultProperty, value); }
        }

        public static readonly DependencyProperty ResultProperty =
            DependencyProperty.Register("Result", typeof(string), typeof(MyViewModel), new UIPropertyMetadata(string.Empty));

        public void Search(string search_clause)
        {
            Result = string.Empty;
            SearchClause = search_clause;
            var worker = new BackgroundWorker();
            worker.DoWork += worker_DoWork;
            worker.RunWorkerCompleted += worker_RunWorkerCompleted;
            IsBusy = true;
            worker.RunWorkerAsync();
        }

        void worker_RunWorkerCompleted(object sender, RunWorkerCompletedEventArgs e)
        {
            IsBusy=false;
            Result = "Sorry, no results found for: " + SearchClause;
        }

        void worker_DoWork(object sender, DoWorkEventArgs e)
        {
            Thread.Sleep(5000);
        }
    }
}

Hope this helps!