Can I buy / purchase stuff using the fingerprint sensor on my iPhone 5s?

Yes, but only from Apple (for now).

Initially, you'll be able to use the iPhone 5s fingerprint sensor to authorize purchases on the iTunes store, through iBooks and the App Store. In other words, in all the places you previously had to enter your iTunes password.

It should be safe to assume that Apple will eventually allow Passbook purchases to be authorized via the fingerprint sensor as well, but no official word on that yet.

There's also no clarity regarding what amount of access, if any, third party apps will have to the fingerprint sensor. What reports there are have been, up to this point, conflicting.

More to come.