Both seem to be used in web development circles, see e.g. HTML5 Cross Browser Polyfills, which says:
So here we're collecting all the shims, fallbacks, and polyfills...
Or, there's the es5-shim project.
In my current project we're using a number of these, and I want to stick them all in the same directory. So, what should I call this directory---shims, or polyfills?
A shim is any piece of code that performs interception of an API call and provides a layer of abstraction. It isn't necessarily restricted to a web application or HTML5/CSS3.
A polyfill is a type of shim that retrofits legacy browsers with modern HTML5/CSS3 features usually using Javascript or Flash.
Answering your specific question, call your directory shims if you want to keep the directory generic.