How to dynamically load and switch the resource file in the web app (ASP.NET) without recompiling?

The question is 6 years old, but I'm still gonna answer it :)

To read .resx files, you need to use System.Resources.ResXResourceReader class from System.Windows.Forms.dll

This is nicely explained here. Just a quick sample for completeness:

using (ResXResourceReader resxReader = new ResXResourceReader(@".\CarResources.resx"))
{
    foreach (DictionaryEntry entry in resxReader) {
        // ...
    } 
}

Sure you can do this easily, and it works for straight XML files. You don't need to use a resx file.

/// <summary>
/// Sets or replaces the ResourceDictionary by dynamically loading
/// a Localization ResourceDictionary from the file path passed in.
/// </summary>
/// <param name="resourceDictionaryFile">The resource dictionary to use to set/replace
/// the ResourceDictionary.</param>
private void SetCultureResourceDictionary(String resourceDictionaryFile)
{
    // Scan all resource dictionaries and remove, if it is string resource distionary
    for ( int index= 0; index < Resources.MergedDictionaries.Count; ++index)
    {
        // Look for an indicator in the resource file that indicates the resource is
        // swappable. For instance in our files the header contains this: 
        // <sys:String x:Key="ResourceDictionaryName">Resources-en-CA</sys:String> 
        if (Resources.MergedDictionaries[index].Contains("ResourceDictionaryName"))
        {
            if ( File.Exists(resourceDictionaryFile) )
            {
                Resources.MergedDictionaries.Remove(Resources.MergedDictionaries[index]);
            }
        }
    }

    // read required resource file to resource dictionary and add to MergedDictionaries collection
    ResourceDictionary newResourceDictionary =  new ResourceDictionary();
    newResourceDictionary.Source = new Uri(resourceDictionaryFile);
    Resources.MergedDictionaries.Add(newResourceDictionary);
}