Does anyone know if I can integrate GrapeJS into my own website so clients could build their own websites using it? IF anyone has done this, how easy is it and are there downsides?
Yes you can.
However it is not straightforward.
If you want to build a Drag Drop Editor like GrapeJS Demo, here is the Source Code - https://github.com/artf/grapesjs-preset-webpage
You can see an implementation at https://codegres.org/dragdrop