If I understand correctly, you'd like a page on GitHub Pages which, when you visit it, within the page there is a virtual machine running PHP and MySQL, and then the same page contains a frame/iframe which accesses the web server running in the virtual machine? I think that might be a think that browsers disallow for security reasons - for one thing I think that having the virtual machine open a listening socket would be a security risk, and having a page hosted on one site have an iframe which accesses localhost might be another security risk, because it would allow your page to access other things running on my host too - isn't this what cross-site scripting attacks are? At least it might be an issue if you try to have the outer page access anything inside the frame using scripting. I could be wrong about that stuff though, I don't know a lot about it.

You might be able to find workarounds for those security settings though.

You might want to look at https://www.pcjs.org/ too, that is also open source and I saw the author talked about hosting at least some part of it on GitHub Pages.