It turns out you can do it. But Circle claims they are not an exchange! You don't "buy" coins on the site. You deposit either fiat or bitcoins and the money is held in your Circle account as bitcoins.
They aren't an exchange. Coinbase is also not an exchange. You trade bitcoins directly with Circle. Exchanges are where you trade with other people.
I know that people will treat it as an exchange and people will have their credit cards and bank accounts closed because they are trying to either trade or arbitrage with circle (just like they do with coinbase and always complain about their bank accounts getting closed)