In addition to what AussieBigfoot
If you don't have the option to start a showdown, it means there's one currently in progress (press RT and there should be flags next to people's gamertags), wait a while for it to finish and you'll be able to start one afterwards.
If you're waiting more than five minutes, it's fair to say the server has probably crashed, so you'll have to exit and try again.
Also if you didn't know, you can switch cars without leaving the car meet. Press
and then "change car".