It fixable through patch against Visual Studio Express config file to redirect assembly binding.
For example to get SQLite designer working in VS C# 2008 Express, you need to add following lines to C:\Program Files\Microsoft Visual Studio 9.0\Common7\IDE\VCSExpress.exe.config on same level with other <dependentAssembly> entries.
<dependentAssembly>
<assemblyIdentity name="Microsoft.VisualStudio.Data" publicKeyToken="b03f5f7f11d50a3a" culture="neutral" />
<bindingRedirect oldVersion="0.0.0.0-9.0.0.0" newVersion="9.0.0.0" />
</dependentAssembly>
For VS C# 2010 you need to add following lines to C:\Program Files\Microsoft Visual Studio 10.0\Common7\IDE\VCSExpress.exe.config
<dependentAssembly>
<assemblyIdentity name="Microsoft.VisualStudio.Data" publicKeyToken="b03f5f7f11d50a3a" culture="neutral" />
<bindingRedirect oldVersion="0.0.0.0-9.0.0.0" newVersion="10.0.0.0" />
</dependentAssembly>
For fix SQLite Designer in Visual Basic Express, change VBExpress.exe.config, for Visual C++ change VCExpress.exe.config respectively.
Although i not recommend it, cause it is against VS Express EULA ("You may not work around any technical limitations").
However seems SQLite design mode works in Visual Studio Web Developer Express 2008 without any modifications.