Not using GoDiagram Web. That would require doing everything on the client, including generating new images “in real time”, which would mean using some sort of plug-in to the browser.
In the Microsoft world that would mean an ActiveX control, which could be a Java applet, Flash, or SVG. Or, it could mean a Windows Forms Control....
Do you want your web application to be integrated into an HTML page? Then you could embed your Windows Forms application Form into your HTML page.
Or do you want it running as a stand-alone application? Use ClickOnce deployment.
In either case you'll be able to re-use your existing code. But it will require dealing with security/permission issues. And it obviously requires having the .NET runtime installed on all of your clients, and that they use IE.