Does the New Yorker magazine ever buy short stories?

I don't have an inside scoop, but think of it this way: They wouldn't waste money hiring employees to read submissions if they had no intention of publishing any of them. They wouldn't have wasted money creating an online portal, either.

Your chances are better if you submit through an agent, not just because agents know how to get the attention of the right editors, but because agents will only submit a manuscript to a magazine if they feel it's a fit for the magazine and if the story is good enough. So once an agent agrees to do this, your story has been vetted by them.

Again, if The New Yorker was unwilling to accept stories directly from unknown writers, they would say so. Since they don't say an agent is required, you can trust that you have a chance and that your submission will be read - at least, the first few lines will. You'll need to hook the reader and keep his/her interest if you want the reader to finish your story and then push to get it into the magazine.

If you can't find out whose short stories the New Yorker has included, then submitting to it would just be a waste of their time. We're supposed to do our own research, even if we want our stories bought for magazines.

Added: No one gets an agent to sell short stories to magazines.

They probably get way too many unpublishable stories than they'd care to get. The acceptance rate of most top-end publishing houses is so tiny that unless your story stands out among the rest (and I mean "really" stands out) the chances of rejection are high. It's a cut-throat market out there.