[disclaimer: I'm an engineer at Continuum working on Wakari]
Wakari Enterprise (http://enterprise.wakari.io) is aiming to do exactly this, and we're hoping to back-port the functionality into Wakari Cloud (http://wakari.io) so "published" IPython Notebooks can have some knobs on them for variable input control, then they can be "invoked" in a sandboxed state, and then the output given back to the user.
However for things that exist now, you should look at Sage Notebook. A few years ago several people worked hard on a Sage Notebook Cell Server that could do exactly what you were asking for: execute small code snippets. I haven't followed it since then, but it seems it is still alive and well from a quick search:
http://sagecell.sagemath.org/?q=ejwwif
http://sagecell.sagemath.org
http://www.sagemath.org/eval.html
For the last URL, check out Graphics->Mandelbrot and you can see that Sage already has some great capabilities for UI widgets that are tied to the "cell execution".