You didn't say what you were looking for - mechanical parts or body and trim. I've found that most common Fiats are pretty cheap when it comes to routine maintenance parts, but get expensive when you start getting into body panels or other model specific parts that aren't in production anymore.
Here's someplace reasonably close (Fiat specialist in South Jersey):
If you're looking for additional parts sources farther afield, here are some of the best (In no particular order):
I've done business with all of these guys, and never had a problem with any of them. Most of them specialize in new parts, but I know Danny O'Donnel at funimported has a large stock of used and secondhand parts as well - he might be a good first place to check, and not too far away in Connecticut.
I also know a guy who's going to be scrapping an 850 Spyder in Reading, PA. It's rough, but depending what you need, he may have it.