<Grid>
    <Grid.ColumnDefinitions>
        <ColumnDefinition Width="*" />
        <ColumnDefinition Width="Auto" />
    </Grid.ColumnDefinitions>
    <TextBox x:Name="textBox" VerticalScrollBarVisibility="Visible" />
    <ScrollBar Grid.Column="1" ValueChanged="ScrollBar_ValueChanged" />
</Grid>

private void ScrollBar_ValueChanged(object sender, RoutedPropertyChangedEventArgs<double> e)
{
    textBox.ScrollToVerticalOffset(e.NewValue);
}