What has made it impossible to buy the tickets? problems with payment? you should be able to use paypal for payments, as to not have a problem with type of card.
What time do you arrive in Rome? Because I also see it being a Sunday, there is a bus at 12.30 and then none until 17.45. SENA/Baltour is the fastest way to get to Siena from Rome, but another option is the bus that connects Siena to Fiumicino (FCO) airport -- you can read more and book online here: http://www.sitbusshuttle.com/en/nuovo-collegamento-siena-aeroporto-di-fiumicino-roma/
Only problem is that there is only one bus in the AM and one in the PM.
If neither work, you'll need to take the train -- and ask your rental to meet you later in the day to get keys, they do it all of the time, you just have to arrange it ahead of time with them so they know to expect you later in the day.
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