My suggestion is to host WebKit.NET in your VB 6 application just like you would any other .NET control.
Although it is a .NET control, which means that it won't work natively out-of-the-box with VB 6, it is possible to use controls developed in .NET with a VB 6 application. Microsoft provides the Interop Forms Toolkit as an interoperability mechanism between the two languages.
Essentially, you'll use one of the .NET languages (it doesn't have to be VB.NET; you could also use C#) to create an ActiveX DLL containing your UserControl
and register it for COM interop. Then, you can add this ActiveX control to your VB 6 project and hook up to handle the events it raises.
You can find a more complete sample on how to do this here on CodeProject, or here on CodeGuru.
Indeed this approach is not going to be trivial to implement. But I suspect that it's your only alternative to the bundled WebBrowser
control, considering that IE was pretty much dominating the browser market back when VB 6 was popular and no one is developing new controls for VB 6 anymore.