Legally, I’m unclear of if an American connecting on a VPN to a live stream in the UK is breaking the law. I would guess that it hasn’t really come up with the TV license people yet.
I don't know about the UK law, but nearby Ireland has a similar "TV Licence". The law there is not "you need a licence to watch TV", but "if you're in Ireland and you have something that's capable of recieving TV signals, then you need a TV licence".
If the UK is similar, then it's not "watch BBC" that requires a licence, but "owning a TV in the UK". The OP almost certainly does not own a TV in the UK, so almost certainly not need a TV licence.
(NB: There may or may not be terms & conditions on the iPlayer website which say you may only access it if you are in UK / have a TV licence / etc., which might make the above the illegal (but not for TV licence reasons). I'm suprised the Olympics works for them, usually BBC iPlayer stuff uses GeoIP to block it from non-UK IP addresses.)