I've been using OpenJSCAD (demo at http://openjscad.org/) for small parts and really like it. With some work it could probably do what you describe but as far as I can tell no one is using it that way.
It would be cool if there was an ecosystem of OpenJSCAD parts that you could import as javascript modules.
I built the URL part out into a little side project. It uses OpenJSCAD but only the OpenSCAD interpreter so it can render the STL for you on the server.
Why are people downvoting this? It seems like a perfectly reasonable wish. If something like this already exists then then please share it. The replies so farention things that are similar in some ways but either aren't parametric or don't have the extensibility that @iamleppert is lining for.
Imagine being able to use npm to share and install parametric parts as easy as:
var Bracket = require('some-bracket');
var L = new Bracket({ ... });