A lot of people use Bespin, but the editor that Mozilla hosts is really bare.
Advantages of running your own server is that you can edit the server for more functionality, like a mini "file browser" instead of loading everything command-line, autocorrection, etc. It's like turning Notepad that has syntax highlighting into an IDE.
As to the original poster...
100% yes, you can get the client on your site.
The server, however, is trickier. Since it's a Python program that's run constantly (I don't know how to describe it. It's like a background process without being in the background), I'd have to say that I doubt djbob will install it for you.
Hopefully when the new server comes out (which has already begun development), you'll be able to run it. The new server is written entirely in Javascript, therefore it won't be a "process".